2016
2016
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Paper 1, Section I, D
Part IA, 2016 commentWhat does it mean to say that a sequence of real numbers converges to ? Suppose that converges to . Show that the sequence given by
also converges to .
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Paper 1, Section I, F
Part IA, 2016 commentLet be the number of pairs of integers such that . What is the radius of convergence of the series ? [You may use the comparison test, provided you state it clearly.]
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Paper 1, Section II, E
Part IA, 2016 commentState the Bolzano-Weierstrass theorem. Use it to show that a continuous function attains a global maximum; that is, there is a real number such that for all .
A function is said to attain a local maximum at if there is some such that whenever . Suppose that is twice differentiable, and that for all . Show that there is at most one at which attains a local maximum.
If there is a constant such that for all , show that attains a global maximum. [Hint: if for all , then is decreasing.]
Must attain a global maximum if we merely require for all Justify your answer.
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Paper 1, Section II, E
Part IA, 2016 commentLet . We say that is a real root of if . Show that if is differentiable and has distinct real roots, then has at least real roots. [Rolle's theorem may be used, provided you state it clearly.]
Let be a polynomial in , where all and . (In other words, the are the nonzero coefficients of the polynomial, arranged in order of increasing power of .) The number of sign changes in the coefficients of is the number of for which . For example, the polynomial has 2 sign changes. Show by induction on that the number of positive real roots of is less than or equal to the number of sign changes in its coefficients.
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Paper 1, Section II, D
Part IA, 2016 commentIf and are sequences converging to and respectively, show that the sequence converges to .
If for all and , show that the sequence converges to .
(a) Find .
(b) Determine whether converges.
Justify your answers.
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Paper 1, Section II, 12F
Part IA, 2016 commentLet satisfy for all .
Show that is continuous and that for all , there exists a piecewise constant function such that
For all integers , let . Show that the sequence converges to 0 .
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Paper 2, Section I, A
Part IA, 2016 comment(a) Find the solution of the differential equation
that is bounded as and satisfies when .
(b) Solve the difference equation
Show that if , the solution that is bounded as and satisfies is approximately .
(c) By setting , explain the relation between parts (a) and (b).
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Paper 2, Section II, A
Part IA, 2016 commentThe function satisfies
What does it mean to say that the point is (i) an ordinary point and (ii) a regular singular point of this differential equation? Explain what is meant by the indicial equation at a regular singular point. What can be said about the nature of the solutions in the neighbourhood of a regular singular point in the different cases that arise according to the values of the roots of the indicial equation?
State the nature of the point of the equation
Set , where , and find the roots of the indicial equation.
(a) Show that one solution of with is
and find a linearly independent solution in the case when is not an integer.
(b) If is a positive integer, show that has a polynomial solution.
(c) What is the form of the general solution of in the case ? [You do not need to find the general solution explicitly.]
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Paper 2, Section , A
Part IA, 2016 comment(a) For each non-negative integer and positive constant , let
By differentiating with respect to , find its value in terms of and .
(b) By making the change of variables , transform the differential equation
into a differential equation for , where .
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Paper 2, Section II, A
Part IA, 2016 comment(a) Find and sketch the solution of
where is the Dirac delta function, subject to and .
(b) A bowl of soup, which Sam has just warmed up, cools down at a rate equal to the product of a constant and the difference between its temperature and the temperature of its surroundings. Initially the soup is at temperature , where .
(i) Write down and solve the differential equation satisfied by .
(ii) At time , when the temperature reaches half of its initial value, Sam quickly adds some hot water to the soup, so the temperature increases instantaneously by , where . Find and for .
(iii) Sketch for .
(iv) Sam wants the soup to be at temperature at time , where . What value of should Sam choose to achieve this? Give your answer in terms of , and .
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Paper 2, Section II,
Part IA, 2016 comment(a) The function satisfies
(i) Define the Wronskian of two linearly independent solutions and . Derive a linear first-order differential equation satisfied by .
(ii) Suppose that is known. Use the Wronskian to write down a first-order differential equation for . Hence express in terms of and .
(b) Verify that is a solution of
where and are constants, provided that these constants satisfy certain conditions which you should determine.
Use the method that you described in part (a) to find a solution which is linearly independent of .
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Paper 2, Section II, A
Part IA, 2016 comment(a) By considering eigenvectors, find the general solution of the equations
and show that it can be written in the form
where and are constants.
(b) For any square matrix , is defined by
Show that if has constant elements, the vector equation has a solution , where is a constant vector. Hence solve and show that your solution is consistent with the result of part (a).
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Paper 4, Section I, B
Part IA, 2016 commentWith the help of definitions or equations of your choice, determine the dimensions, in terms of mass , length , time and charge , of the following quantities:
(i) force;
(ii) moment of a force (i.e. torque);
(iii) energy;
(iv) Newton's gravitational constant ;
(v) electric field ;
(vi) magnetic field ;
(vii) the vacuum permittivity .
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Paper 4, Section , B
Part IA, 2016 commentThe radial equation of motion of a particle moving under the influence of a central force is
where is the angular momentum per unit mass of the particle, is a constant, and is a positive constant.
Show that circular orbits with are possible for any positive value of , and that they are stable to small perturbations that leave unchanged if .
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Paper 4, Section II, B
Part IA, 2016 comment(a) A rocket, moving non-relativistically, has speed and mass at a time after it was fired. It ejects mass with constant speed relative to the rocket. Let the total momentum, at time , of the system (rocket and ejected mass) in the direction of the motion of the rocket be . Explain carefully why can be written in the form
If the rocket experiences no external force, show that
Derive the expression corresponding to for the total kinetic energy of the system at time . Show that kinetic energy is not necessarily conserved.
(b) Explain carefully how should be modified for a rocket moving relativistically, given that there are no external forces. Deduce that
where and hence that
(c) Show that and agree in the limit . Briefly explain the fact that kinetic energy is not conserved for the non-relativistic rocket, but relativistic energy is conserved for the relativistic rocket.
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Paper 4, Section II, B
Part IA, 2016 commentA particle of unit mass moves with angular momentum in an attractive central force field of magnitude , where is the distance from the particle to the centre and is a constant. You may assume that the equation of its orbit can be written in plane polar coordinates in the form
where and is the eccentricity. Show that the energy of the particle is
A comet moves in a parabolic orbit about the Sun. When it is at its perihelion, a distance from the Sun, and moving with speed , it receives an impulse which imparts an additional velocity of magnitude directly away from the Sun. Show that the eccentricity of its new orbit is , and sketch the two orbits on the same axes.
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Paper 4, Section II, B
Part IA, 2016 comment(a) Alice travels at constant speed to Alpha Centauri, which is at distance from Earth. She then turns around (taking very little time to do so), and returns at speed . Bob stays at home. By how much has Bob aged during the journey? By how much has Alice aged? [No justification is required.]
Briefly explain what is meant by the twin paradox in this context. Why is it not a paradox?
(b) Suppose instead that Alice's world line is given by
where is a positive constant. Bob stays at home, at , where . Alice and Bob compare their ages on both occasions when they meet. By how much does Bob age? Show that Alice ages by .
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Paper 4, Section II, B
Part IA, 2016 commentState what the vectors and represent in the following equation:
where is the acceleration due to gravity.
Assume that the radius of the Earth is , that , and that there are seconds in a day. Use these data to determine roughly the order of magnitude of each term on the right hand side of in the case of a particle dropped from a point at height above the surface of the Earth.
Taking again , find the time of the particle's fall in the absence of rotation.
Use a suitable approximation scheme to show that
where is the position vector of the point at which the particle lands, and is the position vector of the point at which the particle would have landed in the absence of rotation.
The particle is dropped at latitude . Find expressions for the approximate northerly and easterly displacements of from in terms of (the magnitudes of and , respectively), and . You should ignore the curvature of the Earth's surface.
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Paper 3, Section I, D
Part IA, 2016 commentLet be a group, and let be a subgroup of . Show that the following are equivalent.
(i) for all .
(ii) is a normal subgroup of and is abelian.
Hence find all abelian quotient groups of the dihedral group of order 10 .
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Paper 3, Section I, D
Part IA, 2016 commentState and prove Lagrange's theorem.
Let be an odd prime number, and let be a finite group of order which has a normal subgroup of order 2 . Show that is a cyclic group.
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Paper 3, Section II, D
Part IA, 2016 commentFor each of the following, either give an example or show that none exists.
(i) A non-abelian group in which every non-trivial element has order
(ii) A non-abelian group in which every non-trivial element has order 3 .
(iii) An element of of order 18 .
(iv) An element of of order 20 .
(v) A finite group which is not isomorphic to a subgroup of an alternating group.
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Paper 3, Section II, D
Part IA, 2016 commentDefine the sign, , of a permutation and prove that it is well defined. Show that the function is a homomorphism.
Show that there is an injective homomorphism such that is non-trivial.
Show that there is an injective homomorphism such that
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Paper 3, Section II,
Part IA, 2016 commentState and prove the orbit-stabiliser theorem.
Let be a prime number, and be a finite group of order with . If is a non-trivial normal subgroup of , show that contains a non-trivial element.
If is a proper subgroup of , show that there is a such that .
[You may use Lagrange's theorem, provided you state it clearly.]
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Paper 3, Section II, D
Part IA, 2016 commentDefine the Möbius group and its action on the Riemann sphere . [You are not required to verify the group axioms.] Show that there is a surjective group homomorphism , and find the kernel of
Show that if a non-trivial element of has finite order, then it fixes precisely two points in . Hence show that any finite abelian subgroup of is either cyclic or isomorphic to .
[You may use standard properties of the Möbius group, provided that you state them clearly.]
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Paper 4, Section I, E
Part IA, 2016 commentFind a pair of integers and satisfying . What is the smallest positive integer congruent to modulo 29 ?
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Paper 4, Section I, E
Part IA, 2016 commentExplain the meaning of the phrase least upper bound; state the least upper bound property of the real numbers. Use the least upper bound property to show that a bounded, increasing sequence of real numbers converges.
Suppose that and that for all . If converges, show that converges.
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Paper 4, Section II, E
Part IA, 2016 comment(a) Let be a set. Show that there is no bijective map from to the power set of . Let for all be the set of sequences with entries in Show that is uncountable.
(b) Let be a finite set with more than one element, and let be a countably infinite set. Determine whether each of the following sets is countable. Justify your answers.
(i) is injective .
(ii) is surjective .
(iii) is bijective .
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Paper 4, Section II,
Part IA, 2016 commentSuppose that and that , where and are relatively prime and greater than 1. Show that there exist unique integers such that and
Now let be the prime factorization of . Deduce that can be written uniquely in the form
where and . Express in this form.
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Paper 4, Section II,
Part IA, 2016 commentState the inclusion-exclusion principle.
Let be a string of digits, where . We say that the string has a run of length if there is some such that either for all or for all . For example, the strings
all have runs of length 3 (underlined), but no run in has length . How many strings of length 6 have a run of length ?
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Paper 4, Section II, 8E
Part IA, 2016 commentDefine the binomial coefficient . Prove directly from your definition that
for any complex number .
(a) Using this formula, or otherwise, show that
(b) By differentiating, or otherwise, evaluate .
Let , where is a non-negative integer. Show that for . Evaluate .
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Paper 2, Section I, F
Part IA, 2016 commentLet be independent random variables, all with uniform distribution on . What is the probability of the event ?
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Paper 2, Section I,
Part IA, 2016 commentDefine the moment-generating function of a random variable . Let be independent and identically distributed random variables with distribution , and let . For , show that
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Paper 2, Section II, F
Part IA, 2016 commentFor any positive integer and positive real number , the Gamma distribution has density defined on by
For any positive integers and , the Beta distribution has density defined on by
Let and be independent random variables with respective distributions and . Show that the random variables and are independent and give their distributions.
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Paper 2, Section II, F
Part IA, 2016 commentWe randomly place balls in bins independently and uniformly. For each with , let be the number of balls in bin .
(a) What is the distribution of ? For , are and independent?
(b) Let be the number of empty bins, the number of bins with two or more balls, and the number of bins with exactly one ball. What are the expectations of and ?
(c) Let , for an integer . What is ? What is the limit of when ?
(d) Instead, let , for an integer . What is ? What is the limit of when ?
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Paper 2, Section II, F
Part IA, 2016 commentLet be a non-negative random variable such that is finite, and let .
(a) Show that
(b) Let and be random variables such that and are finite. State and prove the Cauchy-Schwarz inequality for these two variables.
(c) Show that
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Paper 2, Section II, F
Part IA, 2016 commentA random graph with nodes is drawn by placing an edge with probability between and for all distinct and , independently. A triangle is a set of three distinct nodes that are all connected: there are edges between and , between and and between and .
(a) Let be the number of triangles in this random graph. Compute the maximum value and the expectation of .
(b) State the Markov inequality. Show that if , for some , then when
(c) State the Chebyshev inequality. Show that if is such that when , then when
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Paper 3, Section I, C
Part IA, 2016 commentState the chain rule for the derivative of a composition , where and are smooth
Consider parametrized curves given by
Calculate the tangent vector in terms of and . Given that is a smooth function in the upper half-plane satisfying
deduce that
If , find .
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Paper 3, Section I, C
Part IA, 2016 commentIf and are vectors in , show that defines a rank 2 tensor. For which choices of the vectors and is isotropic?
Write down the most general isotropic tensor of rank 2 .
Prove that defines an isotropic rank 3 tensor.
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Paper 3, Section II, C
Part IA, 2016 commentWhat is a conservative vector field on ?
State Green's theorem in the plane .
(a) Consider a smooth vector field defined on all of which satisfies
By considering
or otherwise, show that is conservative.
(b) Now let . Show that there exists a smooth function such that .
Calculate , where is a smooth curve running from to . Deduce that there does not exist a smooth function which satisfies and which is, in addition, periodic with period 1 in each coordinate direction, i.e. .
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Paper 3, Section II, C
Part IA, 2016 commentDefine the Jacobian of a smooth mapping . Show that if is the vector field with components
then . If is another such mapping, state the chain rule formula for the derivative of the composition , and hence give in terms of and .
Let be a smooth vector field. Let there be given, for each , a smooth mapping such that as . Show that
for some , and express in terms of . Assuming now that , deduce that if then for all . What geometric property of the mapping does this correspond to?
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Paper 3, Section II, C
Part IA, 2016 comment(a) For smooth scalar fields and , derive the identity
and deduce that
Here is the Laplacian, where is the unit outward normal, and is the scalar area element.
(b) Give the expression for in terms of . Hence show that
(c) Assume that if , where and as , then
The vector fields and satisfy
Show that . In the case that , with , show that
and hence that
Verify that given by does indeed satisfy . [It may be useful to make a change of variables in the right hand side of .]
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Paper 3, Section II, C
Part IA, 2016 comment(a) Let
and let be a circle of radius lying in a plane with unit normal vector . Calculate and use this to compute . Explain any orientation conventions which you use.
(b) Let be a smooth vector field such that the matrix with entries is symmetric. Prove that for every circle .
(c) Let , where and let be the circle which is the intersection of the sphere with the plane . Calculate .
(d) Let be the vector field defined, for , by
Show that . Let be the curve which is the intersection of the cylinder with the plane . Calculate .
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Paper 1, Section I, A
Part IA, 2016 commentLet be a solution of
where and . For which values of do the following hold?
(i) .
(ii) .
(iii) .
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Paper 1, Section I, C
Part IA, 2016 commentWrite down the general form of a rotation matrix. Let be a real matrix with positive determinant such that for all . Show that is a rotation matrix.
Let
Show that any real matrix which satisfies can be written as , where is a real number and is a rotation matrix.
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Paper 1, Section II, A
Part IA, 2016 comment(a) Use suffix notation to prove that
(b) Show that the equation of the plane through three non-colinear points with position vectors and is
where is the position vector of a point in this plane.
Find a unit vector normal to the plane in the case and .
(c) Let be the position vector of a point in a given plane. The plane is a distance from the origin and has unit normal vector , where . Write down the equation of this plane.
This plane intersects the sphere with centre at and radius in a circle with centre at and radius . Show that
Find in terms of and . Hence find in terms of and .
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Paper 1, Section II,
Part IA, 2016 commentThe real symmetric matrix has eigenvectors of unit length , with corresponding eigenvalues , where . Prove that the eigenvalues are real and that .
Let be any (real) unit vector. Show that
What can be said about if
Let be the set of all (real) unit vectors of the form
Show that for some . Deduce that
The matrix is obtained by removing the first row and the first column of . Let be the greatest eigenvalue of . Show that
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Paper 1, Section II, B
Part IA, 2016 commentWhat does it mean to say that a matrix can be diagonalised? Given that the real matrix has eigenvectors satisfying , explain how to obtain the diagonal form of . Prove that is indeed diagonal. Obtain, with proof, an expression for the trace of in terms of its eigenvalues.
The elements of are given by
Determine the elements of and hence show that, if is an eigenvalue of , then
Assuming that can be diagonalised, give its diagonal form.
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Paper 1, Section II,
Part IA, 2016 comment(a) Show that the equations
determine and uniquely if and only if .
Write the following system of equations
in matrix form . Use Gaussian elimination to solve the system for , and . State a relationship between the rank and the kernel of a matrix. What is the rank and what is the kernel of ?
For which values of , and is it possible to solve the above system for and ?
(b) Define a unitary matrix. Let be a real symmetric matrix, and let be the identity matrix. Show that for arbitrary , where . Find a similar expression for . Prove that is well-defined and is a unitary matrix.
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Paper 3, Section I, G
Part IB, 2016 comment(a) Let be a subset of . What does it mean to say that a sequence of functions is uniformly convergent?
(b) Which of the following sequences of functions are uniformly convergent? Justify your answers.
(i)
(ii) .
(iii) , .
(iv) .
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Paper 4, Section I, G
Part IB, 2016 comment(a) What does it mean to say that a mapping from a metric space to itself is a contraction?
(b) State carefully the contraction mapping theorem.
(c) Let . By considering the metric space with
or otherwise, show that there exists a unique solution of the system of equations
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Paper 2, Section I, G
Part IB, 2016 comment(a) What does it mean to say that the function is differentiable at the point ? Show from your definition that if is differentiable at , then is continuous at .
(b) Suppose that there are functions such that for every ,
Show that is differentiable at if and only if each is differentiable at .
(c) Let be given by
Determine at which points the function is differentiable.
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Paper 1, Section II, G
Part IB, 2016 commentLet be a metric space.
(a) What does it mean to say that is a Cauchy sequence in ? Show that if is a Cauchy sequence, then it converges if it contains a convergent subsequence.
(b) Let be a Cauchy sequence in .
(i) Show that for every , the sequence converges to some .
(ii) Show that as .
(iii) Let be a subsequence of . If are such that , show that as .
(iv) Show also that for every and ,
(v) Deduce that has a subsequence such that for every and ,
and
(c) Suppose that every closed subset of has the property that every contraction mapping has a fixed point. Prove that is complete.
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Paper 4, Section II, G
Part IB, 2016 comment(a) Let be a real vector space. What does it mean to say that two norms on are Lipschitz equivalent? Prove that every norm on is Lipschitz equivalent to the Euclidean norm. Hence or otherwise, show that any linear map from to is continuous.
(b) Let be a linear map between normed real vector spaces. We say that is bounded if there exists a constant such that for all . Show that is bounded if and only if is continuous.
(c) Let denote the space of sequences of real numbers such that is convergent, with the norm . Let be the sequence with and if . Let be the sequence . Show that the subset is linearly independent. Let be the subspace it spans, and consider the linear map defined by
Is continuous? Justify your answer.
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Paper 3, Section II, G
Part IB, 2016 commentLet be a metric space.
(a) What does it mean to say that a function is uniformly continuous? What does it mean to say that is Lipschitz? Show that if is Lipschitz then it is uniformly continuous. Show also that if is a Cauchy sequence in , and is uniformly continuous, then the sequence is convergent.
(b) Let be continuous, and be sequentially compact. Show that is uniformly continuous. Is necessarily Lipschitz? Justify your answer.
(c) Let be a dense subset of , and let be a continuous function. Show that there exists at most one continuous function such that for all , . Prove that if is uniformly continuous, then such a function exists, and is uniformly continuous.
[A subset is dense if for any nonempty open subset , the intersection is nonempty.]
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Paper 2, Section II, G
Part IB, 2016 comment(a) What is a norm on a real vector space?
(b) Let be the space of linear maps from to . Show that
defines a norm on , and that if then .
(c) Let be the space of real matrices, identified with in the usual way. Let be the subset
Show that is an open subset of which contains the set .
(d) Let be the map . Show carefully that the series converges on to . Hence or otherwise, show that is twice differentiable at 0 , and compute its first and second derivatives there.
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Paper 4, Section I, G
Part IB, 2016 commentState carefully Rouché's theorem. Use it to show that the function has exactly one zero in the quadrant
and that .
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Paper 3, Section II, G
Part IB, 2016 comment(a) Prove Cauchy's theorem for a triangle.
(b) Write down an expression for the winding number of a closed, piecewise continuously differentiable curve about a point which does not lie on .
(c) Let be a domain, and a holomorphic function with no zeroes in . Suppose that for infinitely many positive integers the function has a holomorphic -th root. Show that there exists a holomorphic function such that .
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Paper 2, Section II, A
Part IB, 2016 commentLet for a positive integer . Let be the anticlockwise contour defined by the square with its four vertices at and . Let
Show that is uniformly bounded on the contours as , and hence that as .
Using this result, establish that
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Paper 1, Section II, A
Part IB, 2016 commentLet and let , for real.
(a) Let A be the map defined by , using the principal branch. Show that A maps the region to the left of the parabola on the plane, with the negative real axis removed, into the vertical strip of the plane between the lines and .
(b) Let be the map defined by . Show that maps the vertical strip of the -plane between the lines and into the region inside the unit circle on the -plane, with the part of the negative real axis removed.
(c) Using the results of parts (a) and (b), show that the map C, defined by , maps the region to the left of the parabola on the -plane, including the negative real axis, onto the unit disc on the -plane.
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Paper 1, Section I, A
Part IB, 2016 commentClassify the singularities of the following functions at both and at the point at infinity on the extended complex plane:
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Paper 3, Section I, A
Part IB, 2016 commentThe function has Fourier transform
where is a real constant. Using contour integration, calculate for . [Jordan's lemma and the residue theorem may be used without proof.]
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Paper 4, Section II, A
Part IB, 2016 comment(a) Show that the Laplace transform of the Heaviside step function is
for .
(b) Derive an expression for the Laplace transform of the second derivative of a function in terms of the Laplace transform of and the properties of at .
(c) A bar of length has its end at fixed. The bar is initially at rest and straight. The end at is given a small fixed transverse displacement of magnitude at . You may assume that the transverse displacement of the bar satisfies the wave equation with some wave speed , and so the tranverse displacement is the solution to the problem:
(i) Show that the Laplace transform of , defined as
is given by
(ii) By use of the binomial theorem or otherwise, express as an infinite series.
(iii) Plot the transverse displacement of the midpoint of the bar against time.
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Paper 2, Section I,
Part IB, 2016 comment(a) Derive the integral form of Ampère's law from the differential form of Maxwell's equations with a time-independent magnetic field, and .
(b) Consider two perfectly-conducting concentric thin cylindrical shells of infinite length with axes along the -axis and radii and . Current flows in the positive -direction in each shell. Use Ampère's law to calculate the magnetic field in the three regions: (i) , (ii) and (iii) , where .
(c) If current now flows in the positive -direction in the inner shell and in the negative -direction in the outer shell, calculate the magnetic field in the same three regions.
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Paper 4, Section I, D
Part IB, 2016 comment(a) Starting from Maxwell's equations, show that in a vacuum,
(b) Suppose that where and are real constants.
(i) What are the wavevector and the polarisation? How is related to ?
(ii) Find the magnetic field .
(iii) Compute and interpret the time-averaged value of the Poynting vector, .
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Paper 1, Section II, D
Part IB, 2016 comment(a) From the differential form of Maxwell's equations with and a time-independent electric field, derive the integral form of Gauss's law.
(b) Derive an expression for the electric field around an infinitely long line charge lying along the -axis with charge per unit length . Find the electrostatic potential up to an arbitrary constant.
(c) Now consider the line charge with an ideal earthed conductor filling the region . State the boundary conditions satisfied by and on the surface of the conductor.
(d) Show that the same boundary conditions at are satisfied if the conductor is replaced by a second line charge at with charge per unit length .
(e) Hence or otherwise, returning to the setup in (c), calculate the force per unit length acting on the line charge.
(f) What is the charge per unit area on the surface of the conductor?
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Paper 3, Section II, D
Part IB, 2016 comment(a) State Faraday's law of induction for a moving circuit in a time-dependent magnetic field and define all the terms that appear.
(b) Consider a rectangular circuit DEFG in the plane as shown in the diagram below. There are two rails parallel to the -axis for starting at at and at . A battery provides an electromotive force between and driving current in a positive sense around DEFG. The circuit is completed with a bar resistor of resistance , length and mass that slides without friction on the rails; it connects at and at . The rest of the circuit has no resistance. The circuit is in a constant uniform magnetic field parallel to the -axis.
[In parts (i)-(iv) you can neglect any magnetic field due to current flow.]

(i) Calculate the current in the bar and indicate its direction on a diagram of the circuit.
(ii) Find the force acting on the bar.
(iii) If the initial velocity and position of the bar are respectively and , calculate and for .
(iv) If , find the total energy dissipated in the circuit after and verify that total energy is conserved.
(v) Describe qualitatively the effect of the magnetic field caused by the induced current flowing in the circuit when .
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Paper 2, Section II, D
Part IB, 2016 comment(a) State the covariant form of Maxwell's equations and define all the quantities that appear in these expressions.
(b) Show that is a Lorentz scalar (invariant under Lorentz transformations) and find another Lorentz scalar involving and .
(c) In some inertial frame the electric and magnetic fields are respectively and . Find the electric and magnetic fields, and , in another inertial frame that is related to by the Lorentz transformation,
where is the velocity of in and .
(d) Suppose that and where , and is a real constant. An observer is moving in with velocity parallel to the -axis. What must be for the electric and magnetic fields to appear to the observer to be parallel? Comment on the case .
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Paper 1, Section I, C
Part IB, 2016 commentConsider the flow field in cartesian coordinates given by
where is a constant. Let denote the whole of excluding the axis.
(a) Determine the conditions on and for the flow to be both incompressible and irrotational in .
(b) Calculate the circulation along any closed curve enclosing the axis.
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Paper 2, Section I, C
Part IB, 2016 commentA steady, two-dimensional unidirectional flow of a fluid with dynamic viscosity is set up between two plates at and . The plate at is stationary while the plate at moves with constant speed . The fluid is also subject to a constant pressure gradient . You may assume that the fluid velocity has the form .
(a) State the equation satisfied by and its boundary conditions.
(b) Calculate .
(c) Show that the value of may be chosen to lead to zero viscous shear stress acting on the bottom plate and calculate the resulting flow rate.
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Paper 1, Section II,
Part IB, 2016 comment(a) For a velocity field , show that , where is the flow vorticity.
(b) For a scalar field , show that if , then is constant along the flow streamlines.
(c) State the Euler equations satisfied by an inviscid fluid of constant density subject to conservative body forces.
(i) If the flow is irrotational, show that an exact first integral of the Euler equations may be obtained.
(ii) If the flow is not irrotational, show that although an exact first integral of the Euler equations may not be obtained, a similar quantity is constant along the flow streamlines provided the flow is steady.
(iii) If the flow is now in a frame rotating with steady angular velocity , establish that a similar quantity is constant along the flow streamlines with an extra term due to the centrifugal force when the flow is steady.
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Paper 4, Section II, C
Part IB, 2016 comment(a) Show that for an incompressible fluid, , where is the flow vorticity,
(b) State the equation of motion for an inviscid flow of constant density in a rotating frame subject to gravity. Show that, on Earth, the local vertical component of the centrifugal force is small compared to gravity. Present a scaling argument to justify the linearisation of the Euler equations for sufficiently large rotation rates, and hence deduce the linearised version of the Euler equations in a rapidly rotating frame.
(c) Denoting the rotation rate of the frame as , show that the linearised Euler equations may be manipulated to obtain an equation for the velocity field in the form
(d) Assume that there exist solutions of the form . Show that where the angle is to be determined.
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Paper 3, Section II, C
Part IB, 2016 commentA layer of thickness of a fluid of density is located above a layer of thickness of a fluid of density . The two-fluid system is bounded by two impenetrable surfaces at and and is assumed to be two-dimensional (i.e. independent of ). The fluid is subsequently perturbed, and the interface between the two fluids is denoted .
(a) Assuming irrotational motion in each fluid, state the equations and boundary conditions satisfied by the flow potentials, and .
(b) The interface is perturbed by small-amplitude waves of the form , with . State the equations and boundary conditions satisfied by the linearised system.
(c) Calculate the dispersion relation of the waves relating the frequency to the wavenumber .
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Paper 1, Section I, F
Part IB, 2016 comment(a) Describe the Poincaré disc model for the hyperbolic plane by giving the appropriate Riemannian metric.
(b) Let be some point. Write down an isometry with .
(c) Using the Poincaré disc model, calculate the distance from 0 to re with
(d) Using the Poincaré disc model, calculate the area of a disc centred at a point and of hyperbolic radius .
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Paper 3, Section I,
Part IB, 2016 comment(a) State Euler's formula for a triangulation of a sphere.
(b) A sphere is decomposed into hexagons and pentagons with precisely three edges at each vertex. Determine the number of pentagons.
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Paper 3, Section II, F
Part IB, 2016 comment(a) Define the cross-ratio of four distinct points . Show that the cross-ratio is invariant under Möbius transformations. Express in terms of .
(b) Show that is real if and only if lie on a line or circle in .
(c) Let lie on a circle in , given in anti-clockwise order as depicted.

Show that is a negative real number, and that is a positive real number greater than 1 . Show that . Use this to deduce Ptolemy's relation on lengths of edges and diagonals of the inscribed 4-gon:
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Paper 2, Section II, F
Part IB, 2016 comment(a) Let be a hyperbolic triangle, with the angle at at least . Show that the side has maximal length amongst the three sides of .
[You may use the hyperbolic cosine formula without proof. This states that if and are the lengths of , and respectively, and and are the angles of the triangle at and respectively, then
(b) Given points in the hyperbolic plane, let be any point on the hyperbolic line segment joining to , and let be any point not on the hyperbolic line passing through . Show that
where denotes hyperbolic distance.
(c) The diameter of a hyperbolic triangle is defined to be
Show that the diameter of is equal to the length of its longest side.
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Paper 4, Section II, F
Part IB, 2016 commentLet be a simple curve in parameterised by arc length with for all , and consider the surface of revolution in defined by the parameterisation
(a) Calculate the first and second fundamental forms for . Show that the Gaussian curvature of is given by
(b) Now take . What is the integral of the Gaussian curvature over the surface of revolution determined by and ?
[You may use the Gauss-Bonnet theorem without proof.]
(c) Now suppose has constant curvature , and suppose there are two points such that is a smooth closed embedded surface. Show that is a unit sphere, minus two antipodal points.
[Do not attempt to integrate an expression of the form when . Study the behaviour of the surface at the largest and smallest possible values of .]
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Paper 3, Section I, E
Part IB, 2016 commentLet be a group of order . Define what is meant by a permutation representation of . Using such representations, show is isomorphic to a subgroup of the symmetric group . Assuming is non-abelian simple, show is isomorphic to a subgroup of . Give an example of a permutation representation of whose kernel is .
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Paper 4, Section I,
Part IB, 2016 commentGive the statement and the proof of Eisenstein's criterion. Use this criterion to show is irreducible in where is a prime.
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Paper 2, Section I, E
Part IB, 2016 commentLet be an integral domain.
Define what is meant by the field of fractions of . [You do not need to prove the existence of .]
Suppose that is an injective ring homomorphism from to a field . Show that extends to an injective ring homomorphism .
Give an example of and a ring homomorphism from to a ring such that does not extend to a ring homomorphism .
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Paper 1, Section II, E
Part IB, 2016 comment(a) Let be an ideal of a commutative ring and assume where the are prime ideals. Show that for some .
(b) Show that is a maximal ideal of . Show that the quotient ring is isomorphic to
(c) For , let be the ideal in . Show that is a maximal ideal. Find a maximal ideal of such that for any . Justify your answers.
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Paper 3, Section II, E
Part IB, 2016 comment(a) Define what is meant by an algebraic integer . Show that the ideal
in is generated by a monic irreducible polynomial . Show that , considered as a -module, is freely generated by elements where .
(b) Assume satisfies . Is it true that the ideal (5) in is a prime ideal? Is there a ring homomorphism ? Justify your answers.
(c) Show that the only unit elements of are 1 and . Show that is not a UFD.
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Paper 4, Section II, E
Part IB, 2016 commentLet be a Noetherian ring and let be a finitely generated -module.
(a) Show that every submodule of is finitely generated.
(b) Show that each maximal element of the set
is a prime ideal. [Here, maximal means maximal with respect to inclusion, and
(c) Show that there is a chain of submodules
such that for each the quotient is isomorphic to for some prime ideal .
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Paper 2, Section II, E
Part IB, 2016 comment(a) State Sylow's theorems and give the proof of the second theorem which concerns conjugate subgroups.
(b) Show that there is no simple group of order 351 .
(c) Let be the finite field and let be the multiplicative group of invertible matrices over . Show that every Sylow 3-subgroup of is abelian.
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Paper 4, Section I, F
Part IB, 2016 commentFor which real numbers do the vectors
not form a basis of ? For each such value of , what is the dimension of the subspace of that they span? For each such value of , provide a basis for the spanned subspace, and extend this basis to a basis of .
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Paper 2, Section I, F
Part IB, 2016 commentFind a linear change of coordinates such that the quadratic form
takes the form
for real numbers and .
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Paper 1, Section ,
Part IB, 2016 comment(a) Consider the linear transformation given by the matrix
Find a basis of in which is represented by a diagonal matrix.
(b) Give a list of matrices such that any linear transformation with characteristic polynomial
and minimal polynomial
is similar to one of the matrices on your list. No two distinct matrices on your list should be similar. [No proof is required.]
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Paper 1, Section II, F
Part IB, 2016 commentLet denote the vector space over or of matrices with entries in . Let denote the trace functional, i.e., if , then
(a) Show that Tr is a linear functional.
(b) Show that for .
(c) Show that is unique in the following sense: If is a linear functional such that for each , then is a scalar multiple of the trace functional. If, in addition, , then Tr.
(d) Let be the subspace spanned by matrices of the form for . Show that is the kernel of Tr.
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Paper 4, Section II, F
Part IB, 2016 comment(a) Let be a linear transformation between finite dimensional vector spaces over a field or .
Define the dual map of . Let be the dual map of . Given a subspace , define the annihilator of . Show that and the image of coincide. Conclude that the dimension of the image of is equal to the dimension of the image of . Show that .
(b) Now suppose in addition that are inner product spaces. Define the adjoint of . Let be linear transformations between finite dimensional inner product spaces. Suppose that the image of is equal to the kernel of . Then show that is an isomorphism.
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Paper 3, Section II, F
Part IB, 2016 commentLet be a linear transformation defined on a finite dimensional inner product space over . Recall that is normal if and its adjoint commute. Show that being normal is equivalent to each of the following statements:
(i) where are self-adjoint operators and ;
(ii) there is an orthonormal basis for consisting of eigenvectors of ;
(iii) there is a polynomial with complex coefficients such that .
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Paper 2, Section II, F
Part IB, 2016 commentLet denote the vector space over a field or of matrices with entries in . Given , consider the two linear transformations defined by
(a) Show that .
[For parts (b) and (c), you may assume the analogous result without proof.]
(b) Now let . For , write for the conjugate transpose of , i.e., . For , define the linear transformation by
Show that .
(c) Again let . Let be the set of Hermitian matrices. [Note that is not a vector space over but only over For and , define . Show that is an -linear operator on , and show that as such,
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Paper 4, Section I, H
Part IB, 2016 commentConsider two boxes, labelled and B. Initially, there are no balls in box and balls in box B. Each minute later, one of the balls is chosen uniformly at random and is moved to the opposite box. Let denote the number of balls in box A at time , so that .
(a) Find the transition probabilities of the Markov chain and show that it is reversible in equilibrium.
(b) Find , where is the next time that all balls are again in box .
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Paper 3, Section I, H
Part IB, 2016 commentLet be a Markov chain such that . Prove that
where . [You may use the strong Markov property without proof.]
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Paper 2, Section II, H
Part IB, 2016 comment(a) Prove that every open communicating class of a Markov chain is transient. Prove that every finite transient communicating class is open. Give an example of a Markov chain with an infinite transient closed communicating class.
(b) Consider a Markov chain with state space and transition probabilities given by the matrix
(i) Compute for a fixed .
(ii) Compute for some .
(iii) Show that converges as , and determine the limit.
[Results from lectures can be used without proof if stated carefully.]
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Paper 1, Section II, H
Part IB, 2016 commentLet be a simple symmetric random walk on the integers, starting at .
(a) What does it mean to say that a Markov chain is irreducible? What does it mean to say that an irreducible Markov chain is recurrent? Show that is irreducible and recurrent.
[Hint: You may find it helpful to use the limit
You may also use without proof standard necessary and sufficient conditions for recurrence.]
(b) What does it mean to say that an irreducible Markov chain is positive recurrent? Determine, with proof, whether is positive recurrent.
(c) Let
be the first time the chain returns to the origin. Compute for a fixed number .
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Paper 2, Section I, A
Part IB, 2016 commentUse the method of characteristics to find in the first quadrant , where satisfies
with boundary data .
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Paper 4, Section I, A
Part IB, 2016 commentConsider the function defined by
Calculate the Fourier series representation for the -periodic extension of this function. Hence establish that
and that
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Paper 3, Section , A
Part IB, 2016 commentCalculate the Green's function given by the solution to
where and is the Dirac -function. Use this Green's function to calculate an explicit solution to the boundary value problem
where , and .
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Paper 1, Section II, A
Part IB, 2016 comment(a) Consider the general self-adjoint problem for on :
where is the eigenvalue, and . Prove that eigenfunctions associated with distinct eigenvalues are orthogonal with respect to a particular inner product which you should define carefully.
(b) Consider the problem for given by
(i) Recast this problem into self-adjoint form.
(ii) Calculate the complete set of eigenfunctions and associated eigenvalues for this problem. [Hint: You may find it useful to make the substitution
(iii) Verify that the eigenfunctions associated with distinct eigenvalues are indeed orthogonal.
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Paper 3, Section II, B
Part IB, 2016 comment(a) Show that the Fourier transform of , for , is
stating clearly any properties of the Fourier transform that you use.
[Hint: You may assume that .]
(b) Consider now the Cauchy problem for the diffusion equation in one space dimension, i.e. solving for satisfying:
where is a positive constant and is specified. Consider the following property of a solution:
Property P: If the initial data is positive and it is non-zero only within a bounded region (i.e. there is a constant such that for all , then for any (however small) and (however large) the solution can be non-zero, i.e. the solution can become non-zero arbitrarily far away after an arbitrarily short time.
Does Property P hold for solutions of the diffusion equation? Justify your answer (deriving any expression for the solution that you use).
(c) Consider now the wave equation in one space dimension:
with given initial data and (and is a constant).
Does Property (with and now replaced by and respectively) hold for solutions of the wave equation? Justify your answer again as above.
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Paper 2, Section II, A
Part IB, 2016 commentConsider a bar of length with free ends, subject to longitudinal vibrations. You may assume that the longitudinal displacement of the bar satisfies the wave equation with some wave speed :
for and with boundary conditions:
for . The bar is initially at rest so that
for , with a spatially varying initial longitudinal displacement given by
for , where is a real constant.
(a) Using separation of variables, show that
(b) Determine a periodic function such that this solution may be expressed as
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Paper 4, Section II, B
Part IB, 2016 commentLet be a 2-dimensional region in with boundary . In this question you may assume Green's second identity:
where denotes the outward normal derivative on , and and are suitably regular functions that include the free space Green's function in two dimensions. You may also assume that the free space Green's function for the Laplace equation in two dimensions is given by
(a) State the conditions required on a function for it to be a Dirichlet Green's function for the Laplace operator on . Suppose that on . Show that if is a Dirichlet Green's function for then
(b) Consider the Laplace equation in the quarter space
with boundary conditions
Using the method of images, show that the solution is given by
where
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Paper 3, Section I,
Part IB, 2016 commentLet be a topological space and be a subset. A limit point of is a point such that any open neighbourhood of intersects . Show that is closed if and only if it contains all its limit points. Explain what is meant by the interior Int and the closure of . Show that if is connected, then is connected.
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Paper 2, Section I, E
Part IB, 2016 commentConsider and with their usual topologies.
(a) Show that compact subsets of a Hausdorff topological space are closed. Show that compact subsets of are closed and bounded.
(b) Show that there exists a complete metric space admitting a surjective continuous map .
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Paper 1, Section II, E
Part IB, 2016 commentLet be a prime number. Define what is meant by the -adic metric on . Show that for we have
Show that the sequence , where , converges to some element in (D.
For define . Show that if and if , then
Let and let be the open ball with centre and radius , with respect to the metric . Show that is a closed subset of with respect to the topology induced by .
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Paper 4, Section II, E
Part IB, 2016 comment(a) Let be a topological space. Define what is meant by a quotient of and describe the quotient topology on the quotient space. Give an example in which is Hausdorff but the quotient space is not Hausdorff.
(b) Let be the 2-dimensional torus considered as the quotient , and let be the quotient map.
(i) Let be the open ball in with centre and radius . Show that is an open subset of and show that has infinitely many connected components. Show each connected component is homeomorphic to .
(ii) Let be an irrational number and let be the line given by the equation . Show that is dense in but .
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Paper 1, Section I, D
Part IB, 2016 comment(a) What are real orthogonal polynomials defined with respect to an inner product What does it mean for such polynomials to be monic?
(b) Real monic orthogonal polynomials, , of degree , are defined with respect to the inner product,
where is a positive weight function. Show that such polynomials obey the three-term recurrence relation,
for appropriate and which should be given in terms of inner products.
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Paper 4, Section I, D
Part IB, 2016 comment(a) Define the linear stability domain for a numerical method to solve . What is meant by an A-stable method?
(b) A two-stage Runge-Kutta scheme is given by
where is the step size and . Show that the order of this scheme is at least two. For this scheme, find the intersection of the linear stability domain with the real axis. Hence show that this method is not A-stable.
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Paper 1, Section II, D
Part IB, 2016 comment(a) Consider a method for numerically solving an ordinary differential equation (ODE) for an initial value problem, . What does it mean for a method to converge over where ? What is the definition of the order of a method?
(b) A general multistep method for the numerical solution of an ODE is
where is a fixed positive integer. Show that this method is at least of order if and only if
(c) State the Dahlquist equivalence theorem regarding the convergence of a multistep method.
(d) Consider the multistep method,
Determine the values of and (in terms of the real parameter ) such that the method is at least third order. For what values of does the method converge?
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Paper 3, Section II, D
Part IB, 2016 comment(a) Determine real quadratic functions such that the interpolation formula,
is exact when is any real polynomial of degree 2 .
(b) Use this formula to construct approximations for and which are exact when is any real polynomial of degree 2 . Calculate these approximations for and comment on your answers.
(c) State the Peano kernel theorem and define the Peano kernel . Use this theorem to find the minimum values of the constants and such that
and
where . Check that these inequalities hold for .
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Paper 2, Section II, D
Part IB, 2016 comment(a) Define a Givens rotation and show that it is an orthogonal matrix.
(b) Define a QR factorization of a matrix with . Explain how Givens rotations can be used to find and .
(c) Let
(i) Find a QR factorization of using Givens rotations.
(ii) Hence find the vector which minimises , where is the Euclidean norm. What is ?
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Paper 1, Section I, H
Part IB, 2016 commentbe the payoff of a two-person zero-sum game, where player I (randomly) picks a row to maximise the expected payoff and player II picks a column to minimise the expected payoff. Find each player's optimal strategy and the value of the game.
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Paper 2, Section I, H
Part IB, 2016 commentUse the simplex algorithm to find the optimal solution to the linear program:
Write down the dual problem and find its solution.
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Paper 4, Section II, H
Part IB, 2016 comment(a) What is the maximal flow problem in a network? Explain the Ford-Fulkerson algorithm. Prove that this algorithm terminates if the initial flow is set to zero and all arc capacities are rational numbers.
(b) Let be an matrix. We say that is doubly stochastic if for and
We say that is a permutation matrix if for all and
for all there exists a unique such that ,
for all there exists a unique such that .
Let be the set of all doubly stochastic matrices. Show that a matrix is an extreme point of if and only if is a permutation matrix.
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Paper 3, Section II, H
Part IB, 2016 comment(a) State and prove the Lagrangian sufficiency theorem.
(b) Let be a given constant, and consider the problem:
Find, with proof, constants such that the optimal solution is given by
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Paper 4, Section I, B
Part IB, 2016 comment(a) Define the quantum orbital angular momentum operator in three dimensions, in terms of the position and momentum operators.
(b) Show that . [You may assume that the position and momentum operators satisfy the canonical commutation relations.]
(c) Let . Show that commutes with .
[In this part of the question you may additionally assume without proof the permuted relations and
[Hint: It may be useful to consider the expression for suitable operators and .]
(d) Suppose that and are normalised eigenstates of with eigenvalues and respectively. Consider the wavefunction
with being a positive constant. Find the earliest time such that the expectation value of in is zero.
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Paper 3, Section I, B
Part IB, 2016 comment(a) Consider a quantum particle moving in one space dimension, in a timeindependent real potential . For a wavefunction , define the probability density and probability current and show that
(b) Suppose now that and , where and are real positive constants, and is a complex constant. Compute the probability current for this wavefunction. Interpret the terms in and comment on how this relates to the computed expression for the probability current.
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Paper 1, Section II, B
Part IB, 2016 comment(a) A particle of mass in one space dimension is confined to move in a potential given by
The normalised initial wavefunction of the particle at time is
(i) Find the expectation value of the energy at time .
(ii) Find the wavefunction of the particle at time .
[Hint: It may be useful to recall the identity .]
(b) The right hand wall of the potential is lowered to a finite constant value giving the new potential:
This potential is set up in the laboratory but the value of is unknown. The stationary states of the potential are investigated and it is found that there exists exactly one bound state. Show that the value of must satisfy
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Paper 3, Section II, B
Part IB, 2016 commentThe spherically symmetric bound state wavefunctions for the Coulomb potential are normalisable solutions of the equation
Here and is the energy of the state.
(a) By writing the wavefunction as , for a suitable constant that you should determine, show that there are normalisable wavefunctions only for energies of the form
with being a positive integer.
(b) The energies in (a) reproduce the predictions of the Bohr model of the hydrogen atom. How do the wavefunctions above compare to the assumptions in the Bohr model?
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Paper 2, Section II, B
Part IB, 2016 commentThe one dimensional quantum harmonic oscillator has Hamiltonian
where and are real positive constants and and are the standard position and momentum operators satisfying the commutation relation . Consider the operators
(a) Show that
(b) Suppose that is an eigenfunction of with eigenvalue . Show that is then also an eigenfunction of and that its corresponding eigenvalue is .
(c) Show that for any normalisable wavefunctions and ,
[You may assume that the operators and are Hermitian.]
(d) With as in (b), obtain an expression for the norm of in terms of and the norm of . [The squared norm of any wavefunction is .]
(e) Show that all eigenvalues of are non-negative.
(f) Using the above results, deduce that each eigenvalue of must be of the form for some non-negative integer .
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Paper 1, Section I, H
Part IB, 2016 commentLet be independent samples from the exponential distribution with density for , where is an unknown parameter. Find the critical region of the most powerful test of size for the hypotheses versus . Determine whether or not this test is uniformly most powerful for testing versus .
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Paper 2, Section I, H
Part IB, 2016 commentThe efficacy of a new medicine was tested as follows. Fifty patients were given the medicine, and another fifty patients were given a placebo. A week later, the number of patients who got better, stayed the same, or got worse was recorded, as summarised in this table:
\begin{tabular}{|l|c|c|} \hline & medicine & placebo \ better & 28 & 22 \ same & 4 & 16 \ worse & 18 & 12 \ \hline \end{tabular}
Conduct a Pearson chi-squared test of size of the hypothesis that the medicine and the placebo have the same effect.
[Hint: You may find the following values relevant:
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Paper 4, Section II, H
Part IB, 2016 commentConsider the linear regression model
for , where the non-zero numbers are known and are such that , the independent random variables have the distribution, and the parameters and are unknown.
(a) Let be the maximum likelihood estimator of . Prove that for each , the random variables and are uncorrelated. Using standard facts about the multivariate normal distribution, prove that and are independent.
(b) Find the critical region of the generalised likelihood ratio test of size for testing versus . Prove that the power function of this test is of the form for some function . [You are not required to find explicitly.]
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Paper 1, Section II, H
Part IB, 2016 comment(a) What does it mean to say a statistic is sufficient for an unknown parameter ? State the factorisation criterion for sufficiency and prove it in the discrete case.
(b) State and prove the Rao-Blackwell theorem.
(c) Let be independent samples from the uniform distribution on for an unknown positive parameter . Consider the two-dimensional statistic
Prove that is sufficient for . Determine, with proof, whether or not is minimally sufficient.
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Paper 3, Section II, H
Part IB, 2016 commentLet be independent samples from the Poisson distribution with mean .
(a) Compute the maximum likelihood estimator of . Is this estimator biased?
(b) Under the assumption that is very large, use the central limit theorem to find an approximate confidence interval for . [You may use the notation for the number such that for a standard normal
(c) Now suppose the parameter has the prior distribution. What is the posterior distribution? What is the Bayes point estimator for for the quadratic loss function Let be another independent sample from the same distribution. Given , what is the posterior probability that ?
[Hint: The density of the distribution is , for .]
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Paper 1, Section I, C
Part IB, 2016 comment(a) Consider the function , where is a real constant. For what values of is the function convex?
(b) In the case , calculate the extremum of on the set of points where
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Paper 3, Section I, C
Part IB, 2016 commentTwo points and are located on the curved surface of the circular cylinder of radius with axis along the -axis. We denote their locations by and using cylindrical polar coordinates and assume . A path is drawn on the cylinder to join and . Show that the path of minimum distance between the points and is a helix, and determine its pitch. [For a helix with axis parallel to the axis, the pitch is the change in after one complete helical turn.]
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Paper 2, Section II, C
Part IB, 2016 commentA flexible wire filament is described by the curve in cartesian coordinates for . The filament is assumed to be almost straight and thus we assume and everywhere.
(a) Show that the total length of the filament is approximately where
(b) Under a uniform external axial force, , the filament adopts the shape which minimises the total energy, , where is the bending energy given by
and where and are -dependent bending rigidities (both known and strictly positive). The filament satisfies the boundary conditions
Derive the Euler-Lagrange equations for and .
(c) In the case where and , show that below a critical force, , which should be determined, the only energy-minimising solution for the filament is straight , but that a new nonzero solution is admissible at .
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Paper 4, Section II, C
Part IB, 2016 commentA fish swims in the ocean along a straight line with speed . The fish starts its journey from rest (zero velocity at ) and, during a given time , swims subject to the constraint that the total distance travelled is . The energy cost for swimming is per unit time, where are known and .
(a) Derive the Euler-Lagrange condition on for the journey to have minimum energetic cost.
(b) In the case solve for assuming that the fish starts at with zero acceleration (in addition to zero velocity).
(c) In the case , the fish can decide between three different boundary conditions for its journey. In addition to starting with zero velocity, it can:
(1) start at with zero acceleration;
(2) end at with zero velocity; or
(3) end at with zero acceleration.
Which of or (3) is the best minimal-energy cost strategy?
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Paper 3, Section II, H
Part II, 2016 comment(a) Let be an affine variety. Define the tangent space of at a point . Say what it means for the variety to be singular at .
Define the dimension of in terms of (i) the tangent spaces of , and (ii) Krull dimension.
(b) Consider the ideal generated by the set . What is
Using the generators of the ideal, calculate the tangent space of a point in . What has gone wrong? [A complete argument is not necessary.]
(c) Calculate the dimension of the tangent space at each point for , and determine the location of the singularities of
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Paper 2, Section II, H
Part II, 2016 commentIn this question we work over an algebraically closed field of characteristic zero. Let and let be the closure of in
(a) Show that is a non-singular curve.
(b) Show that is a regular differential on .
(c) Compute the divisor of . What is the genus of ?
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Paper 4, Section II, H
Part II, 2016 comment(a) Let be a smooth projective curve, and let be an effective divisor on . Explain how defines a morphism from to some projective space.
State a necessary and sufficient condition on so that the pull-back of a hyperplane via is an element of the linear system .
State necessary and sufficient conditions for to be an isomorphism onto its image.
(b) Let now have genus 2 , and let be an effective canonical divisor. Show that the morphism is a morphism of degree 2 from to .
Consider the divisor for points with . Show that the linear system associated to this divisor induces a morphism from to a quartic curve in . Show furthermore that , with , if and only if .
[You may assume the Riemann-Roch theorem.]
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Paper 1, Section II, H
Part II, 2016 commentLet be an algebraically closed field.
(a) Let and be affine varieties defined over . Given a map , define what it means for to be a morphism of affine varieties.
(b) Let be the map given by
Show that is a morphism. Show that the image of is a closed subvariety of and determine its ideal.
(c) Let be the map given by
Show that the image of is a closed subvariety of .
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Paper 3, Section II, G
Part II, 2016 commentConstruct a space as follows. Let each be homeomorphic to the standard 2-sphere . For each , let be the North pole and let be the South pole . Then
where for each (and indices are taken modulo 3 ).
(a) Describe the universal cover of .
(b) Compute the fundamental group of (giving your answer as a well-known group).
(c) Show that is not homotopy equivalent to the circle .
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Paper 2, Section II, G
Part II, 2016 comment(a) Let be simplicial complexes, and a continuous map. What does it mean to say that is a simplicial approximation to
(b) Define the barycentric subdivision of a simplicial complex , and state the Simplicial Approximation Theorem.
(c) Show that if is a simplicial approximation to then .
(d) Show that the natural inclusion induces a surjective map on fundamental groups.
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Paper 1, Section II, G
Part II, 2016 commentLet be the 2-dimensional torus. Let be the inclusion of the coordinate circle , and let be the result of attaching a 2-cell along .
(a) Write down a presentation for the fundamental group of (with respect to some basepoint), and identify it with a well-known group.
(b) Compute the simplicial homology of any triangulation of .
(c) Show that is not homotopy equivalent to any compact surface.
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Paper 4, Section II, G
Part II, 2016 commentLet be the 2-dimensional torus, and let be constructed from by removing a small open disc.
(a) Show that is homotopy equivalent to .
(b) Show that the universal cover of is homotopy equivalent to a tree.
(c) Exhibit (finite) cell complexes , such that and are not homotopy equivalent but their universal covers are.
[State carefully any results from the course that you use.]
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Paper 1, Section II, A
Part II, 2016 commentA particle in one dimension of mass and energy is incident from on a potential with as and for . The relevant solution of the time-independent Schrödinger equation has the asymptotic form
Explain briefly why a pole in the reflection amplitude at with corresponds to the existence of a stable bound state in this potential. Indicate why a pole in just below the real -axis, at with , corresponds to a quasi-stable bound state. Find an approximate expression for the lifetime of such a quasi-stable state.
Now suppose that
where and are constants. Compute the reflection amplitude in this case and deduce that there are quasi-stable bound states if is large. Give expressions for the wavefunctions and energies of these states and compute their lifetimes, working to leading non-vanishing order in for each expression.
[ You may assume for and .]
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Paper 4, Section II, A
Part II, 2016 commentLet be a Bravais lattice. Define the dual lattice and show that
obeys for all , where are constants. Suppose is the potential for a particle of mass moving in a two-dimensional crystal that contains a very large number of lattice sites of and occupies an area . Adopting periodic boundary conditions, plane-wave states can be chosen such that
The allowed wavevectors are closely spaced and include all vectors in . Find an expression for the matrix element in terms of the coefficients . [You need not discuss additional details of the boundary conditions.]
Now suppose that , where is a dimensionless constant. Find the energy for a particle with wavevector to order in non-degenerate perturbation theory. Show that this expansion in breaks down on the Bragg lines in k-space defined by the condition
and explain briefly, without additional calculations, the significance of this for energy levels in the crystal.
Consider the particular case in which has primitive vectors
where and are orthogonal unit vectors. Determine the polygonal region in -space corresponding to the lowest allowed energy band.
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Paper 2, Section II, A
Part II, 2016 commentA particle of mass moves in three dimensions subject to a potential localised near the origin. The wavefunction for a scattering process with incident particle of wavevector is denoted . With reference to the asymptotic form of , define the scattering amplitude , where is the wavevector of the outgoing particle with .
By recasting the Schrödinger equation for as an integral equation, show that
[You may assume that
is the Green's function for which obeys the appropriate boundary conditions for a scattering solution.]
Now suppose , where is a dimensionless constant. Determine the first two non-zero terms in the expansion of in powers of , giving each term explicitly as an integral over one or more position variables
Evaluate the contribution to of order in the case , expressing the answer as a function of and the scattering angle (defined so that .
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Paper 3, Section II, J
Part II, 2016 comment(a) State the thinning and superposition properties of a Poisson process on . Prove the superposition property.
(b) A bi-infinite Poisson process with is a process with independent and stationary increments over . Moreover, for all , the increment has the Poisson distribution with parameter . Prove that such a process exists.
(c) Let be a bi-infinite Poisson process on of intensity . We identify it with the set of points of discontinuity of , i.e., . Show that if we shift all the points of by the same constant , then the resulting process is also a Poisson process of intensity .
Now suppose we shift every point of by or with equal probability. Show that the final collection of points is still a Poisson process of intensity . [You may assume the thinning and superposition properties for the bi-infinite Poisson process.]
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Paper 3, Section II, A
Part II, 2016 comment(a) A spinless charged particle moves in an electromagnetic field defined by vector and scalar potentials and . The wavefunction for the particle satisfies the time-dependent Schrödinger equation with Hamiltonian
Consider a gauge transformation
for some function . Define covariant derivatives with respect to space and time, and show that satisfies the Schrödinger equation with potentials and .
(b) Suppose that in part (a) the magnetic field has the form , where is a constant, and that . Find a suitable with and determine the energy levels of the Hamiltonian when the -component of the momentum of the particle is zero. Suppose in addition that the particle is constrained to lie in a rectangular region of area in the -plane. By imposing periodic boundary conditions in the -direction, estimate the degeneracy of each energy level. [You may use without proof results for a quantum harmonic oscillator, provided they are clearly stated.]
(c) An electron is a charged particle of spin with a two-component wavefunction governed by the Hamiltonian
where is the unit matrix and denotes the Pauli matrices. Find the energy levels for an electron in the constant magnetic field defined in part (b), assuming as before that the -component of the momentum of the particle is zero.
Consider such electrons confined to the rectangular region defined in part (b). Ignoring interactions between the electrons, show that the ground state energy of this system vanishes for less than some integer which you should determine. Find the ground state energy for , where is a positive integer.
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Paper 2, Section II, J
Part II, 2016 comment(a) Define an queue and write without proof its stationary distribution. State Burke's theorem for an queue.
(b) Let be an queue with arrival rate and service rate started from the stationary distribution. For each , denote by the last time before that a customer departed the queue and the first time after that a customer departed the queue. If there is no arrival before time , then we set . What is the limit as of ? Explain.
(c) Consider a system of queues serving a finite number of customers in the following way: at station , customers are served immediately and the service times are independent exponentially distributed with parameter ; after service, each customer goes to station with probability . We assume here that the system is closed, i.e., for all .
Let be the state space of the Markov chain. Write down its -matrix. Also write down the -matrix corresponding to the position in the network of one customer (that is, when ). Show that there is a unique distribution such that . Show that
defines an invariant measure for the chain. Are the queue lengths independent at equilibrium?
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Paper 4, Section II, J
Part II, 2016 comment(a) Give the definition of a renewal process. Let be a renewal process associated with with . Show that almost surely
(b) Give the definition of Kingman's -coalescent. Let be the first time that all blocks have coalesced. Find an expression for . Let be the total length of the branches of the tree, i.e., if is the first time there are lineages, then . Show that as . Show also that for all , where is a positive constant, and that in probability
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Paper 1, Section II, J
Part II, 2016 comment(a) Define a continuous-time Markov chain with infinitesimal generator and jump chain .
(b) Prove that if a state is transient for , then it is transient for .
(c) Prove or provide a counterexample to the following: if is positive recurrent for , then it is positive recurrent for .
(d) Consider the continuous-time Markov chain on with non-zero transition rates given by
Determine whether is transient or recurrent. Let , where is the first jump time. Does have an invariant distribution? Justify your answer. Calculate .
(e) Let be a continuous-time random walk on with and for all with . Determine for which values of the walk is transient and for which it is recurrent. In the recurrent case, determine the range of for which it is also positive recurrent. [Here denotes the Euclidean norm of .]
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Paper 3, Section II, C
Part II, 2016 commentConsider the integral
for real , where . Find and sketch, in the complex -plane, the paths of steepest descent through the endpoints and and through any saddle point(s). Obtain the leading order term in the asymptotic expansion of for large positive . What is the order of the next term in the expansion? Justify your answer.
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Paper 2, Section II, C
Part II, 2016 commentWhat is meant by the asymptotic relation
Show that
and find the corresponding result in the sector .
What is meant by the asymptotic expansion
Show that the coefficients are determined uniquely by . Show that if is analytic at , then its Taylor series is an asymptotic expansion for as for any .
Show that
defines a solution of the equation for any smooth and rapidly decreasing function . Use the method of stationary phase to calculate the leading-order behaviour of as , for fixed .
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Paper 4, Section II, C
Part II, 2016 commentConsider the equation
where is a small parameter and is smooth. Search for solutions of the form
and, by equating powers of , obtain a collection of equations for the which is formally equivalent to (1). By solving explicitly for and derive the Liouville- Green approximate solutions to (1).
For the case , where and is a positive constant, consider the eigenvalue problem
Show that any eigenvalue is necessarily positive. Solve the eigenvalue problem exactly when .
Obtain Liouville-Green approximate eigenfunctions for (2) with , and give the corresponding Liouville Green approximation to the eigenvalues . Compare your results to the exact eigenvalues and eigenfunctions in the case , and comment on this.
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Paper 4, Section I,
Part II, 2016 comment(a) Construct a register machine to compute the function . State the relationship between partial recursive functions and partial computable functions. Show that the function is partial recursive.
(b) State Rice's theorem. Show that the set is recursively enumerable but not recursive.
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Paper 3, Section I,
Part II, 2016 comment(a) Define what it means for a context-free grammar (CFG) to be in Chomsky normal form (CNF). Can a CFG in CNF ever define a language containing ? If denotes the result of converting an arbitrary CFG into one in CNF, state the relationship between and .
(b) Let be a CFG in CNF, and let be a word of length . Show that every derivation of in requires precisely steps. Using this, give an algorithm that, on input of any word on the terminals of , decides if or not.
(c) Convert the following CFG into a grammar in CNF:
Does in this case? Justify your answer.
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Paper 2, Section ,
Part II, 2016 comment(a) Which of the following are regular languages? Justify your answers.
(i) is a nonempty string of alternating 's and 's .
(ii) .
(b) Write down a nondeterministic finite-state automaton with -transitions which accepts the language given by the regular expression . Describe in words what this language is.
(c) Is the following language regular? Justify your answer.
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Paper 1, Section I,
Part II, 2016 commentState the pumping lemma for context-free languages (CFLs). Which of the following are CFLs? Justify your answers.
(i) .
(ii) .
(iii) is a prime .
Let be CFLs. Show that is also a CFL.
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Paper 3, Section II, F
Part II, 2016 comment(a) Let be a deterministic finite-state automaton. Define the extended transition function , and the language accepted by , denoted . Let , and . Prove that .
(b) Let where , and let .
(i) Show that there exist such that , where we interpret as if .
(ii) Show that .
(iii) Show that for all .
(c) Prove the following version of the pumping lemma. Suppose , with . Then can be broken up into three words such that , and for all , the word is also in .
(d) Hence show that
(i) if contains a word of length at least , then it contains infinitely many words;
(ii) if , then it contains a word of length less than ;
(iii) if contains all words in of length less than , then .
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Paper 1, Section II, F
Part II, 2016 comment(a) Define a recursive set and a recursively enumerable (r.e.) set. Prove that is recursive if and only if both and are r.e.
(b) Define the halting set . Prove that is r.e. but not recursive.
(c) Let be r.e. sets. Prove that and are r.e. Show by an example that the union of infinitely many r.e. sets need not be r.e.
(d) Let be a recursive set and a (total) recursive function. Prove that the set is r.e. Is it necessarily recursive? Justify your answer.
[Any use of Church's thesis in your answer should be explicitly stated.]
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Paper 4, Section I, E
Part II, 2016 commentUsing conservation of angular momentum in the body frame, derive the Euler equations for a rigid body:
[You may use the formula without proof.]
Assume that the principal moments of inertia satisfy . Determine whether a rotation about the principal 3-axis leads to stable or unstable perturbations.
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Paper 1, Section I, E
Part II, 2016 commentConsider a one-parameter family of transformations such that for all time , and
where is a Lagrangian and a dot denotes differentiation with respect to . State and prove Noether's theorem.
Consider the Lagrangian
where the potential is a function of two variables. Find a continuous symmetry of this Lagrangian and construct the corresponding conserved quantity. Use the Euler-Lagrange equations to explicitly verify that the function you have constructed is independent of .
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Paper 2, Section I, E
Part II, 2016 commentConsider the Lagrangian
where are positive constants and is a positive integer. Find three conserved quantities and show that satisfies
where is a polynomial of degree which should be determined.
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Paper 3, Section I, E
Part II, 2016 commentConsider a six-dimensional phase space with coordinates for . Compute the Poisson brackets , where .
Consider the Hamiltonian
and show that the resulting Hamiltonian system admits three Poisson-commuting independent first integrals.
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Paper 2, Section II, E
Part II, 2016 commentDefine what it means for the transformation given by
to be canonical. Show that a transformation is canonical if and only if
Show that the transformation given by
is canonical for any real constant . Find the corresponding generating function.
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Paper 4, Section II,
Part II, 2016 commentA particle of unit mass is attached to one end of a light, stiff rod of length . The other end of the rod is held at a fixed position, such that the rod is free to swing in any direction. Write down the Lagrangian for the system giving a clear definition of any angular variables you introduce. [You should assume the acceleration is constant.]
Find two independent constants of the motion.
The particle is projected horizontally with speed from a point where the rod lies at an angle to the downward vertical, with . In terms of and , find the critical speed such that the particle always remains at its initial height.
The particle is now projected horizontally with speed but from a point at angle to the vertical, where . Show that the height of the particle oscillates, and find the period of oscillation in terms of and .
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Paper 1, Section I, G
Part II, 2016 commentFind the average length of an optimum decipherable binary code for a source that emits five words with probabilities
Show that, if a source emits words (with ), and if are the lengths of the codewords in an optimum encoding over the binary alphabet, then
[You may assume that an optimum encoding can be given by a Huffman encoding.]
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Paper 2, Section I, G
Part II, 2016 commentShow that the binary channel with channel matrix
has capacity .
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Paper 3, Section I,
Part II, 2016 commentDescribe in words the unicity distance of a cryptosystem.
Denote the cryptosystem by , in the usual way, and let and be random variables and . The unicity distance is formally defined to be the least such that . Derive the formula
where , and is the alphabet of the ciphertext. Make clear any assumptions you make.
The redundancy of a language is given by . If a language has zero redundancy what is the unicity of any cryptosystem?
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Paper 4, Section I, G
Part II, 2016 commentDescribe the Rabin-Williams scheme for coding a message as modulo a certain . Show that, if is chosen appropriately, breaking this code is equivalent to factorising the product of two primes.
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Paper 1, Section II, G
Part II, 2016 commentWhat does it mean to say a binary code has length , size and minimum distance d?
Let be the largest value of for which there exists an -code. Prove that
where
Another bound for is the Singleton bound given by
Prove the Singleton bound and give an example of a linear code of length that satisfies .
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Paper 2, Section II, G
Part II, 2016 commentDefine a code of length , where is odd, over the field of 2 elements with design distance . Show that the minimum weight of such a code is at least . [Results about the Vandermonde determinant may be quoted without proof, provided they are stated clearly.]
Let be a root of . Let be the code of length 15 with defining set . Find the generator polynomial of and the rank of . Determine the error positions of the following received words:
(i) ,
(ii) .
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Paper 1, Section I, C
Part II, 2016 commentThe expansion scale factor, , for an isotropic and spatially homogeneous universe containing material with pressure and mass density obeys the equations
where the speed of light is set equal to unity, is Newton's constant, is a constant equal to 0 or , and is the cosmological constant. Explain briefly the interpretation of these equations.
Show that these equations imply
Hence show that a static solution with constant exists when if
What must the value of be, if the density is non-zero?
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Paper 2, Section I, C
Part II, 2016 commentA spherical cloud of mass has radius and initial radius . It contains material with uniform mass density , and zero pressure. Ignoring the cosmological constant, show that if it is initially at rest at and the subsequent gravitational collapse is governed by Newton's law , then
Suppose is given parametrically by
where at . Derive a relation between and and hence show that the cloud collapses to radius at
where is the initial mass density of the cloud.
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Paper 3, Section I, C
Part II, 2016 commentA universe contains baryonic matter with background density and density inhomogeneity , together with non-baryonic dark matter with background density and density inhomogeneity . After the epoch of radiation-matter density equality, , the background dynamics are governed by
where is the Hubble parameter.
The dark-matter density is much greater than the baryonic density and so the time-evolution of the coupled density perturbations, at any point , is described by the equations
Show that
where and are independent of time. Neglecting modes in and that decay with increasing time, show that the baryonic density inhomogeneity approaches
where is independent of time.
Briefly comment on the significance of your calculation for the growth of baryonic density inhomogeneities in the early universe.
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Paper 4, Section I, C
Part II, 2016 commentThe external gravitational potential due to a thin spherical shell of radius and mass per unit area , centred at , will equal the gravitational potential due to a point mass at , at any distance , provided
where depends on the radius of the shell. For which values of does this equation have solutions of the form , where is constant? Evaluate in each case and find the relation between the mass of the shell and .
Hence show that the general gravitational force
has a potential satisfying . What is the cosmological significance of the constant ?
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Paper 3, Section II, C
Part II, 2016 commentThe early universe is described by equations (with units such that )
where . The universe contains only a self-interacting scalar field with interaction potential so that the density and pressure are given by
Show that
Explain the slow-roll approximation and apply it to equations (1) and (2) to show that it leads to
If with a positive constant and , show that
and that, for small , the scale factor expands to leading order in as
Comment on the relevance of this result for inflationary cosmology.
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Paper 1, Section II, C
Part II, 2016 commentThe distribution function gives the number of particles in the universe with position in and momentum in at time . It satisfies the boundary condition that as and as . Its evolution obeys the Boltzmann equation
where the collision term describes any particle production and annihilation that occurs.
The universe expands isotropically and homogeneously with expansion scale factor , so the momenta evolve isotropically with magnitude . Show that the Boltzmann equation simplifies to
The number densities of particles and of antiparticles are defined in terms of their distribution functions and , and momenta and , by
and the collision term may be assumed to be of the form
where determines the annihilation cross-section of particles by antiparticles and is the production rate of particles.
By integrating equation with respect to the momentum and assuming that is a constant, show that
where . Assuming the same production rate for antiparticles, write down the corresponding equation satisfied by and show that
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Paper 3, Section II, G
Part II, 2016 commentExplain what it means for an embedded surface in to be minimal. What is meant by an isothermal parametrization of an embedded surface ? Prove that if is isothermal then is minimal if and only if the components of are harmonic functions on . [You may assume the formula for the mean curvature of a parametrized embedded surface,
where (respectively ) are the coefficients of the first (respectively second) fundamental forms.]
Let be an embedded connected minimal surface in which is closed as a subset of , and let be a plane which is disjoint from . Assuming that local isothermal parametrizations always exist, show that if the Euclidean distance between and is attained at some point , i.e. , then is a plane parallel to 川.
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Paper 4, Section II, G
Part II, 2016 commentFor a smooth embedded surface, define what is meant by a geodesic curve on . Show that any geodesic curve has constant speed .
For any point , show that there is a parametrization of some open neighbourhood of in , with having coordinates , for which the first fundamental form is
for some strictly positive smooth function on . State a formula for the Gaussian curvature of in in terms of . If on , show that is a function of only, and that we may reparametrize so that the metric is locally of the form , for appropriate local coordinates .
[You may assume that for any and nonzero , there exists (for some a unique geodesic with and , and that such geodesics depend smoothly on the initial conditions and
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Paper 2, Section II, G
Part II, 2016 commentIf an embedded surface contains a line , show that the Gaussian curvature is non-positive at each point of . Give an example where the Gaussian curvature is zero at each point of .
Consider the helicoid given as the image of in under the map
What is the image of the corresponding Gauss map? Show that the Gaussian curvature at a point is given by , and hence is strictly negative everywhere. Show moreover that there is a line in passing through any point of .
[General results concerning the first and second fundamental forms on an oriented embedded surface and the Gauss map may be used without proof in this question.]
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Paper 1, Section II, G
Part II, 2016 commentDefine what is meant by the regular values and critical values of a smooth map of manifolds. State the Preimage Theorem and Sard's Theorem.
Suppose now that . If is compact, prove that the set of regular values is open in , but show that this may not be the case if is non-compact.
Construct an example with and compact for which the set of critical values is not a submanifold of .
[Hint: You may find it helpful to consider the case and . Properties of bump functions and the function may be assumed in this question.]
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Paper 3, Section II, E
Part II, 2016 commentConsider the dependence of the system
on the parameter . Find the fixed points and plot their location in the -plane. Hence, or otherwise, deduce that there are bifurcations at and .
Investigate the bifurcation at by making the substitutions and . Find the extended centre manifold in the form correct to second order. Find the evolution equation on the extended centre manifold to second order, and determine the stability of its fixed points.
Use a plot to show which branches of fixed points in the -plane are stable and which are unstable, and state, without calculation, the type of bifurcation at . Hence sketch the structure of the phase plane very close to the origin for in the cases (i) and (ii) .
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Paper 1, Section II, E
Part II, 2016 commentConsider the dynamical system
where . Find and classify the fixed points of the system.
Use Dulac's criterion with a weighting function of the form and a suitable choice of to show that there are no periodic orbits for . For the case use the same weighting function to find a function which is constant on trajectories. [Hint: is Hamiltonian.]
Calculate the stable manifold at correct to quadratic order in .
Sketch the phase plane for the cases (i) and (ii) .
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Paper 4, Section II, E
Part II, 2016 commentConsider the map defined on by
and let be the open interval . Explain what it means for to have a horseshoe on by identifying the relevant intervals in the definition.
Let . Show that .
Find the sets and .
Consider the ternary (base-3) representation of numbers in . Show that
where the function of the ternary digits should be identified. What is the ternary representation of the non-zero fixed point? What do the ternary representations of elements of have in common?
Show that has sensitive dependence on initial conditions on , that is topologically transitive on , and that periodic points are dense in . [Hint: You may assume that for .]
Briefly state the relevance of this example to the relationship between Glendinning's and Devaney's definitions of chaos.
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Paper 2, Section II, E
Part II, 2016 commentConsider the nonlinear oscillator
(a) Use the Hamiltonian for to find a constraint on the size of the domain of stability of the origin when .
(b) Assume that given there exists an such that all trajectories eventually remain within the region . Show that there must be a limit cycle, stating carefully any result that you use. [You need not show that there is only one periodic orbit.]
(c) Use the energy-balance method to find the approximate amplitude of the limit cycle for .
(d) Find the approximate shape of the limit cycle for , and calculate the leading-order approximation to its period.
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Paper 1, Section II, E
Part II, 2016 commentA point particle of charge and mass moves in an electromagnetic field with 4 -vector potential , where is position in spacetime. Consider the action
where is an arbitrary parameter along the particle's worldline and is the Minkowski metric.
(a) By varying the action with respect to , with fixed endpoints, obtain the equation of motion
where is the proper time, is the velocity 4-vector, and is the field strength tensor.
(b) This particle moves in the field generated by a second point charge that is held at rest at the origin of some inertial frame. By choosing a suitable expression for and expressing the first particle's spatial position in spherical polar coordinates , show from the action that
are constants, where and overdots denote differentiation with respect to .
(c) Show that when the motion is in the plane ,
Hence show that the particle's orbit is bounded if , and that the particle can reach the origin in finite proper time if .
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Paper 3, Section II, E
Part II, 2016 commentThe current density in an antenna lying along the -axis takes the form
where is a constant and . Show that at distances for which both and , the retarded vector potential in Lorenz gauge is
where and . Evaluate the integral to show that
In the far-field, where , the electric and magnetic fields are given by
By calculating the Poynting vector, show that the time-averaged power radiated per unit solid angle is
[You may assume that in Lorenz gauge, the retarded potential due to a localised current distribution is
where the retarded time
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Paper 4, Section II, E
Part II, 2016 comment(a) A uniform, isotropic dielectric medium occupies the half-space . The region is in vacuum. State the boundary conditions that should be imposed on and at .
(b) A linearly polarized electromagnetic plane wave, with magnetic field in the -plane, is incident on the dielectric from . The wavevector makes an acute angle to the normal . If the dielectric has frequency-independent relative permittivity , show that the fraction of the incident power that is reflected is
where , and the angle should be specified. [You should ignore any magnetic response of the dielectric.]
(c) Now suppose that the dielectric moves at speed along the -axis, the incident angle , and the magnetic field of the incident radiation is along the -direction. Show that the reflected radiation propagates normal to the surface , has the same frequency as the incident radiation, and has magnetic field also along the -direction. [Hint: You may assume that under a standard Lorentz boost with speed along the -direction, the electric and magnetic field components transform as
where .]
(d) Show that the fraction of the incident power reflected from the moving dielectric
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Paper 4 , Section II, B
Part II, 2016 commentA thin layer of fluid of viscosity occupies the gap between a rigid flat plate at and a flexible no-slip boundary at . The flat plate moves with constant velocity and the flexible boundary moves with no component of velocity in the -direction.
State the two-dimensional lubrication equations governing the dynamics of the thin layer of fluid. Given a pressure gradient , solve for the velocity profile in the fluid and calculate the flux . Deduce that the pressure gradient satisfies
The shape of the flexible boundary is a periodic travelling wave, i.e. and , where and are constants. There is no applied average pressure gradient, so the pressure is also periodic with . Show that
where denotes the average over a period. Calculate the shear stress on the plate.
The speed is such that there is no need to apply an external tangential force to the plate in order to maintain its motion. Show that
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Paper 3, Section II, B
Part II, 2016 commentA cylindrical pipe of radius and length contains two viscous fluids arranged axisymmetrically with fluid 1 of viscosity occupying the central region , where , and fluid 2 of viscosity occupying the surrounding annular region . The flow in each fluid is assumed to be steady and unidirectional, with velocities and respectively, with respect to cylindrical coordinates aligned with the pipe. A fixed pressure drop is applied between the ends of the pipe.
Starting from the Navier-Stokes equations, derive the equations satisfied by and , and state all the boundary conditions. Show that the pressure gradient is constant.
Solve for the velocity profile in each fluid and calculate the corresponding flow rates, and .
Derive the relationship between and that yields the same flow rate in each fluid. Comment on the behaviour of in the limits and , illustrating your comment by sketching the flow profiles.
Hint: In cylindrical coordinates ,
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Paper 2, Section II, B
Part II, 2016 commentFor a two-dimensional flow in plane polar coordinates , state the relationship between the streamfunction and the flow components and . Show that the vorticity is given by , and deduce that the streamfunction for a steady two-dimensional Stokes flow satisfies the biharmonic equation
A rigid stationary circular disk of radius occupies the region . The flow far from the disk tends to a steady straining flow , where is a constant. Inertial forces may be neglected. Calculate the streamfunction, , for the far-field flow.
By making an appropriate assumption about its dependence on , find the streamfunction for the flow around the disk, and deduce the flow components, and .
Calculate the tangential surface stress, , acting on the boundary of the disk.
Hints: In plane polar coordinates ,
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Paper 1, Section II, B
Part II, 2016 commentState the vorticity equation and interpret the meaning of each term.
A planar vortex sheet is diffusing in the presence of a perpendicular straining flow. The flow is everywhere of the form , where as , and and are constants. Show that the vorticity has the form , and obtain a scalar equation describing the evolution of .
Explain physically why the solution approaches a steady state in which the vorticity is concentrated near . Use scaling to estimate the thickness of the steady vorticity layer as a function of and the kinematic viscosity .
Determine the steady vorticity profile, , and the steady velocity profile, .
Hint:
State, with a brief physical justification, why you might expect this steady flow to be unstable to long-wavelength perturbations, defining what you mean by long.
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Paper 4, Section I, 7A
Part II, 2016 commentConsider the equation for :
State necessary and sufficient conditions on and for to be (i) an ordinary point or (ii) a regular singular point. Derive the corresponding conditions for the point .
Determine the most general equation of the form that has regular singular points at and , with all other points being ordinary.
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Paper 3, Section I, A
Part II, 2016 commentThe functions and have Laplace transforms and respectively, and for . The convolution of and is defined by
Express the Laplace transform of in terms of and .
Now suppose that and for , where . Find expressions for and by using a standard integral formula for the Gamma function. Find an expression for by using a standard integral formula for the Beta function. Hence deduce that
for all .
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Paper 1, Section I, A
Part II, 2016 commentEvaluate the integral
where is a real number, for (i) and (ii) .
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Paper 2, Section I, A
Part II, 2016 commentThe Euler product formula for the Gamma function is
Use this to show that
where is a constant, independent of . Find the value of .
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Paper 2, Section II, A
Part II, 2016 commentThe Hurwitz zeta function is defined for by
State without proof the complex values of for which this series converges.
Consider the integral
where is the Hankel contour. Show that provides an analytic continuation of the Hurwitz zeta function for all . Include in your account a careful discussion of removable singularities. [Hint: .]
Show that has a simple pole at and find its residue.
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Paper 1, Section II, A
Part II, 2016 comment(a) Legendre's equation for is
Let be a closed contour. Show by direct substitution that for within
is a non-trivial solution of Legendre's equation.
(b) Now consider
for real and . The closed contour is defined to start at the origin, wind around in a counter-clockwise direction, then wind around in a clockwise direction, then return to the origin, without encircling the point . Assuming that does not lie on the real interval , show by deforming onto this interval that functions may be defined as limits of with .
Find an explicit expression for and verify that it satisfies Legendre's equation with .
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Paper 2, Section II, H
Part II, 2016 comment(a) Let be a finite separable field extension. Show that there exist only finitely many intermediate fields .
(b) Define what is meant by a normal extension. Is a normal extension? Justify your answer.
(c) Prove Artin's lemma, which states: if is a field extension, is a finite subgroup of , and is the fixed field of , then is a Galois extension with .
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Paper 3, Section II, H
Part II, 2016 comment(a) Let be the 13 th cyclotomic extension of , and let be a 13 th primitive root of unity. What is the minimal polynomial of over ? What is the Galois group ? Put . Show that is a Galois extension and find .
(b) Define what is meant by a Kummer extension. Let be a field of characteristic zero and let be the th cyclotomic extension of . Show that there is a sequence of Kummer extensions such that is contained in .
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Paper 1, Section II, H
Part II, 2016 comment(a) Prove that if is a field and , then there exists a splitting field of over . [You do not need to show uniqueness of .]
(b) Let and be algebraically closed fields of the same characteristic. Show that either is isomorphic to a subfield of or is isomorphic to a subfield of . [For subfields of and field homomorphisms with , 2, we say if is a subfield of and . You may assume the existence of a maximal pair with respect to the partial order just defined.]
(c) Give an example of a finite field extension such that there exist where is separable over but is not separable over
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Paper 4, Section II, H
Part II, 2016 comment(a) Let and let be the splitting field of over . Show that is isomorphic to . Let be a root of . Show that is neither a radical extension nor a solvable extension.
(b) Let and let be the splitting field of over . Is it true that has an element of order 29 ? Justify your answer. Using reduction mod techniques, or otherwise, show that has an element of order 3 .
[Standard results from the course may be used provided they are clearly stated.]
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Paper 4, Section II, D
Part II, 2016 commentA spherically symmetric static spacetime has metric
where is a positive constant, and units such that are used.
(a) Explain why a time-like geodesic may be assumed, without loss of generality, to lie in the equatorial plane . For such a geodesic, show that the quantities
are constants of the motion, where a dot denotes differentiation with respect to proper time, . Hence find a first-order differential equation for .
(b) Consider a massive particle fired from the origin, . Show that the particle will return to the origin and find the proper time taken.
(c) Show that circular orbits are possible for any and determine whether such orbits are stable. Show that on any such orbit a clock measures coordinate time.
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Paper 1, Section II, D
Part II, 2016 commentConsider a family of geodesics with an affine parameter and the tangent vector on each curve. The equation of geodesic deviation for a vector field is
where denotes the directional covariant derivative .
(i) Show that if
then satisfies .
(ii) Show that and satisfy .
(iii) Show that if is a Killing vector field, meaning that , then satisfies .
(iv) Show that if satisfies , where is a scalar field and is a time-like unit vector field, then
[You may use: for any vector field
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Paper 2, Section II, D
Part II, 2016 commentThe Kasner (vacuum) cosmological model is defined by the line element
where are constants with and . Show that .
Write down four equations that determine the null geodesics of the Kasner model.
If is the tangent vector to the trajectory of a photon and is the four-velocity of a comoving observer (i.e., an observer at rest in the coordinate system above), what is the physical interpretation of ?
Let be a comoving observer at the origin, , and let be a comoving source of photons located on one of the spatial coordinate axes.
(i) Show that photons emitted by and observed by can be either redshifted or blue-shifted, depending on the location of .
(ii) Given any fixed time , show that there are locations for on each coordinate axis from which no photons reach for .
Now suppose that and . Does the property in (ii) still hold?
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Paper 3, Section II, D
Part II, 2016 commentFor a spacetime that is nearly flat, the metric can be expressed in the form
where is a flat metric (not necessarily diagonal) with constant components, and the components of and their derivatives are small. Show that
where indices are raised and lowered using .
[You may assume that
For the line element
where and its derivatives are small, show that the linearised vacuum field equations reduce to , where is the two-dimensional Laplacian operator in and .
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Paper 3, Section II, G
Part II, 2016 commentDefine the chromatic polynomial of a graph . Show that if has vertices and edges then
where and for all . [You may assume the deletion-contraction relation, provided that you state it clearly.]
Show that for every graph (with ) we have . Show also that if and only if is disconnected.
Explain why is not the chromatic polynomial of any graph.
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Paper 2, Section II, G
Part II, 2016 commentDefine the Turán graph , where and are positive integers with . For which and is regular? For which and does contain as a subgraph?
State and prove Turán's theorem.
Let be unit vectors in the plane. Prove that the number of pairs for which has length less than 1 is at most .
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Paper 4, Section II, G
Part II, 2016 commentState Menger's theorem in both the vertex form and the edge form. Explain briefly how the edge form of Menger's theorem may be deduced from the vertex form.
(a) Show that if is 3 -connected then contains a cycle of even length.
(b) Let be a connected graph with all degrees even. Prove that is even. [Hint: if is a minimal set of edges whose removal disconnects , let be a component of and consider the degrees of the vertices of in the graph .] Give an example to show that can be odd.
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Paper 1, Section II, 16G
Part II, 2016 comment(a) Show that if is a planar graph then . [You may assume Euler's formula, provided that you state it precisely.]
(b) (i) Prove that if is a triangle-free planar graph then .
(ii) Prove that if is a planar graph of girth at least 6 then .
(iii) Does there exist a constant such that, if is a planar graph of girth at least , then Justify your answer.
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Paper 3, Section II, D
Part II, 2016 commentWhat is meant by an auto-Bäcklund transformation?
The sine-Gordon equation in light-cone coordinates is
where and is to be understood modulo . Show that the pair of equations
constitute an auto-Bäcklund transformation for (1).
By noting that is a solution to (1), use the transformation (2) to derive the soliton (or 'kink') solution to the sine-Gordon equation. Show that this solution can be expressed as
for appropriate constants and .
[Hint: You may use the fact that const.]
The following function is a solution to the sine-Gordon equation:
Verify that this represents two solitons travelling towards each other at the same speed by considering constant and taking an appropriate limit.
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Paper 1, Section II, D
Part II, 2016 commentWhat does it mean for an evolution equation to be in Hamiltonian form? Define the associated Poisson bracket.
An evolution equation is said to be bi-Hamiltonian if it can be written in Hamiltonian form in two distinct ways, i.e.
for Hamiltonian operators and functionals . By considering the sequence defined by the recurrence relation
show that bi-Hamiltonian systems possess infinitely many first integrals in involution. [You may assume that can always be solved for , given .]
The Harry Dym equation for the function is
This equation can be written in Hamiltonian form with
Show that the Harry Dym equation possesses infinitely many first integrals in involution. [You need not verify the Jacobi identity if your argument involves a Hamiltonian operator.]
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Paper 2, Section II, D
Part II, 2016 commentWhat does it mean for to describe a 1-parameter group of transformations? Explain how to compute the vector field
that generates such a 1-parameter group of transformations.
Suppose now . Define the th prolongation, , of and the vector field which generates it. If is defined by show that
where and are functions to be determined.
The curvature of the curve in the -plane is given by
Rotations in the -plane are generated by the vector field
Show that the curvature at a point along a plane curve is invariant under such rotations. Find two further transformations that leave invariant.
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Paper 3, Section II, I
Part II, 2016 comment(a) Define Banach spaces and Euclidean spaces over . [You may assume the definitions of vector spaces and inner products.]
(b) Let be the space of sequences of real numbers with finitely many non-zero entries. Does there exist a norm on such that is a Banach space? Does there exist a norm such that is Euclidean? Justify your answers.
(c) Let be a normed vector space over satisfying the parallelogram law
for all . Show that is an inner product on . [You may use without proof the fact that the vector space operations and are continuous with respect to . To verify the identity , you may find it helpful to consider the parallelogram law for the pairs and
(d) Let be an incomplete normed vector space over which is not a Euclidean space, and let be its dual space with the dual norm. Is a Banach space? Is it a Euclidean space? Justify your answers.
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Paper 2, Section II, I
Part II, 2016 comment(a) Let be a topological space and let denote the normed vector space of bounded continuous real-valued functions on with the norm . Define the terms uniformly bounded, equicontinuous and relatively compact as applied to subsets .
(b) The Arzela-Ascoli theorem [which you need not prove] states in particular that if is compact and is uniformly bounded and equicontinuous, then is relatively compact. Show by examples that each of the compactness of , uniform boundedness of , and equicontinuity of are necessary conditions for this conclusion.
(c) Let be a topological space. Assume that there exists a sequence of compact subsets of such that and . Suppose is uniformly bounded and equicontinuous and moreover satisfies the condition that, for every , there exists such that for every and for every . Show that is relatively compact.
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Paper 1, Section II, I
Part II, 2016 comment(a) State the closed graph theorem.
(b) Prove the closed graph theorem assuming the inverse mapping theorem.
(c) Let be Banach spaces and be linear maps. Suppose that is bounded and is both bounded and injective. Show that is bounded.
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Paper 4, Section II, I
Part II, 2016 commentLet be a complex Hilbert space.
(a) Let be a bounded linear map. Show that the spectrum of is a subset of .
(b) Let be a bounded self-adjoint linear map. For , let and . If , show that .
(c) Let be a compact self-adjoint linear map. For , show that is finite-dimensional.
(d) Let be a closed, proper, non-trivial subspace. Let be the orthogonal projection to .
(i) Prove that is self-adjoint.
(ii) Determine the spectrum and the point spectrum of .
(iii) Find a necessary and sufficient condition on for to be compact.
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Paper 2, Section II, F
Part II, 2016 commentDefine the von Neumann hierarchy of sets , and show that each is a transitive set. Explain what is meant by saying that a binary relation on a set is well-founded and extensional. State Mostowski's Theorem.
Let be the binary relation on defined by: if and only if appears in the base-2 expansion of (i.e., the unique expression for as a sum of distinct powers of 2 ). Show that is well-founded and extensional. To which transitive set is isomorphic? Justify your answer.
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Paper 3, Section II, F
Part II, 2016 commentState the Completeness Theorem for the first-order predicate calculus, and deduce the Compactness Theorem.
Let be a first-order theory over a signature whose axioms all have the form where is a (possibly empty) string of variables and is quantifier-free. Show that every substructure of a -model is a -model, and deduce that if is consistent then it has a model in which every element is the interpretation of a closed term of . You may assume the result that if is a substructure of and is a quantifier-free formula with free variables, then .]
Now suppose where is a quantifier-free formula with one free variable . Show that there is a finite list of closed terms of such that
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Paper 1, Section II, F
Part II, 2016 commentWhich of the following statements are true? Justify your answers. (a) Every ordinal is of the form , where is a limit ordinal and . (b) Every ordinal is of the form , where is an ordinal and . (c) If , then for some . (d) If , then is uncountable. (e) If and , then is uncountable.
[Standard laws of ordinal arithmetic may be assumed, but if you use the Division Algorithm you should prove it.]
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Paper 4, Section II, F
Part II, 2016 comment(a) State Zorn's Lemma, and use it to prove that every nontrivial distributive lattice admits a lattice homomorphism .
(b) Let be a propositional theory in a given language . Sketch the way in which the equivalence classes of formulae of , modulo -provable equivalence, may be made into a Boolean algebra. [Detailed proofs are not required, but you should define the equivalence relation explicitly.]
(c) Hence show how the Completeness Theorem for propositional logic may be deduced from the result of part (a).
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Paper 4, Section I, B
Part II, 2016 commentA stochastic birth-death process is given by the master equation
where is the probability that there are individuals in the population at time for and for . Give a brief interpretation of and .
Derive an equation for , where is the generating function
Now assume that . Show that at steady state
and find the corresponding mean and variance.
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Paper 3, Section , B
Part II, 2016 commentA delay model for a population of size at discrete time is given by
Show that for there is a non-trivial equilibrium, and analyse its stability. Show that, as is increased from 1 , the equilibrium loses stability at and find the approximate periodicity close to equilibrium at this point.
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Paper 2, Section I, B
Part II, 2016 comment(a) The populations of two competing species satisfy
where and . Sketch the phase diagram (limiting attention to .
The relative abundance of species 1 is defined by . Show that
where is a constant that should be determined.
(b) Consider the spatial system
and consider a travelling-wave solution of the form representing one species invading territory previously occupied by another species . By linearising near the front of the invasion, show that the wave speed is given by .
[You may assume that the solution to the full nonlinear system will settle to the slowest possible linear wave speed.]
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Paper 1, Section I, B
Part II, 2016 commentConsider an epidemic model where susceptibles are vaccinated at per capita rate , but immunity (from infection or vaccination) is lost at per capita rate . The system is given by
where are the susceptibles, are the infecteds, is the total population size and all parameters are positive. The basic reproduction ratio satisfies .
Find the critical vaccination rate , in terms of and , such that the system has an equilibrium with the disease present if . Show that this equilibrium is stable when it exists.
Find the long-term outcome for and if .
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Paper 3, Section II, B
Part II, 2016 commentThe Fitzhugh-Nagumo model is given by
where and .
For , by considering the nullclines in the -plane, show that there is a unique equilibrium. Sketch the phase diagram
At the system is at the equilibrium, and is then 'switched on' to be for , where is a constant. Describe carefully how suitable choices of can represent a system analogous to (i) a neuron firing once, and (ii) a neuron firing repeatedly. Illustrate your answer with phase diagrams and also plots of against for each case. Find the threshold for that separates these cases. Comment briefly from a biological perspective on the behaviour of the system when and .
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Paper 4, Section II, B
Part II, 2016 commentThe population densities of two types of cell are given by and . The system is described by the equations
where and are positive constants.
(a) Identify the terms which involve interaction between the cell types, and briefly describe what each of these terms might represent.
(b) Consider the system without spatial dynamics. Find the condition on for there to be a non-trivial spatially homogeneous solution that is stable to spatially invariant disturbances.
(c) Consider now the full spatial system, and consider small spatial perturbations proportional to of the solution found in part (b). Show that for sufficiently large (the precise threshold should be found) the spatially homogeneous solution is stable to perturbations with either small or large wavenumber, but is unstable to perturbations at some intermediate wavenumber.
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Paper 2, Section II, F
Part II, 2016 comment(a) Prove that is a fundamental unit in . [You may not assume the continued fraction algorithm.]
(b) Determine the ideal class group of .
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Paper 1, Section II, F
Part II, 2016 comment(a) Let be an irreducible polynomial of degree a root of , and . Show that .
(b) Now suppose . Write down the matrix representing multiplication by with respect to the basis for . Hence show that
(c) Suppose . Determine . [You may quote any standard result, as long as you state it clearly.]
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Paper 4, Section II, F
Part II, 2016 commentLet be a number field, and a prime in . Explain what it means for to be inert, to split completely, and to be ramified in .
(a) Show that if and for some , then 2 does not split completely in .
(b) Let , with and square-free. Determine, in terms of , whether splits completely, is inert, or ramifies in . Hence show that the primes which ramify in are exactly those which divide .
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Paper 3, Section I, I
Part II, 2016 commentShow that the exact power of a prime dividing is . By considering the prime factorisation of , show that
Setting , deduce that for sufficiently large
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Paper 4 , Section I, I
Part II, 2016 commentCompute the continued fraction expansion of , and use it to find two solutions to where and are positive integers.
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Paper 2, Section I, I
Part II, 2016 commentDefine the Legendre symbol and the Jacobi symbol. Compute the Jacobi symbols and , stating clearly any properties of these symbols that you use.
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Paper 4, Section II, 10I
Part II, 2016 comment(a) Define Euler's totient function and show that .
(b) State Lagrange's theorem concerning roots of polynomials mod .
(c) Let be a prime. Proving any results you need about primitive roots, show that has exactly roots.
(d) Show that if and are both primes then is a Fermat pseudoprime for precisely a third of all bases.
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Paper 1, Section I, I
Part II, 2016 commentDefine the Riemann zeta function for . State and prove the alternative formula for as an Euler product. Hence or otherwise show that for .
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Paper 3, Section II, I
Part II, 2016 commentWhat does it mean for a positive definite binary quadratic form to be reduced?
Prove that every positive definite binary quadratic form is equivalent to a reduced form, and that there are only finitely many reduced forms with given discriminant.
State a criterion for a positive integer to be represented by a positive definite binary quadratic form with discriminant , and hence determine which primes are represented by .
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Paper 4, Section II, B
Part II, 2016 comment(a) Describe an implementation of the power method for determining the eigenvalue of largest modulus and its associated eigenvector for a matrix that has a unique eigenvalue of largest modulus.
Now let be a real matrix with distinct eigenvalues satisfying and . The power method is applied to , with an initial condition such that , where is the eigenvector associated with . Show that the power method does not converge. Explain why and become linearly dependent as .
(b) Consider the following variant of the power method, called the two-stage power method, applied to the matrix of part (a):
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Pick satisfying . Let . Set and .
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Calculate and calculate that minimise
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If , solve and let the roots be and . They are accepted as eigenvalues of , and the corresponding eigenvectors are estimated as and .
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Otherwise, divide and by the current value of , increase by 1 and return to Step .
Explain the justification behind Step 2 of the algorithm.
(c) Let , and suppose that, for a large value of , the two-stage power method yields the vectors
Find two eigenvalues of and the corresponding eigenvectors.
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Paper 2, Section II, B
Part II, 2016 comment(a) The advection equation
is discretised using an equidistant grid with stepsizes and . The spatial derivatives are approximated with central differences and the resulting ODEs are approximated with the trapezoidal rule. Write down the relevant difference equation for determining from . What is the name of this scheme? What is the local truncation error?
The boundary condition is periodic, . Explain briefly how to write the discretised scheme in the form , where the matrices and , to be identified, have a circulant form. Using matrix analysis, find the range of for which the scheme is stable. [Standard results may be used without proof if quoted carefully.]
[Hint: An circulant matrix has the form
All such matrices have the same set of eigenvectors , where , and the corresponding eigenvalues are .]
(b) Consider the advection equation on the unit square
where satisfies doubly periodic boundary conditions, , and and are given doubly periodic functions. The system is discretised with the Crank-Nicolson scheme, with central differences for the space derivatives, using an equidistant grid with stepsizes and . Write down the relevant difference equation, and show how to write the scheme in the form
where the matrix should be identified. Describe how (*) can be approximated by Strang splitting, and explain the advantages of doing so.
[Hint: Inversion of the matrix in part (a) has a similar computational cost to that of a tridiagonal matrix.]
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Paper 1, Section II, B
Part II, 2016 comment(a) Consider the periodic function
on the interval . The -point discrete Fourier transform of is defined by
where and . Compute , using the Fast Fourier Transform (FFT). Compare your result with what you get by computing directly from . Carefully explain all your computations.
(b) Now let be any analytic function on that is periodic with period 1 . The continuous Fourier transform of is defined by
Use integration by parts to show that the Fourier coefficients decay spectrally.
Explain what it means for the discrete Fourier transform of to approximate the continuous Fourier transform with spectral accuracy. Prove that it does so.
What can you say about the behaviour of as for fixed ?
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Paper 3, Section II, B
Part II, 2016 comment(a) Define the Jacobi and Gauss-Seidel iteration schemes for solving a linear system of the form , where and , giving formulae for the corresponding iteration matrices and in terms of the usual decomposition .
Show that both iteration schemes converge when results from discretization of Poisson's equation on a square with the five-point formula, that is when
and . [You may state the Householder-John theorem without proof.]
(b) For the matrix given in :
(i) Calculate the eigenvalues of and deduce its spectral radius .
(ii) Show that each eigenvector of is related to an eigenvector of by a transformation of the form for a suitable value of .
(iii) Deduce that . What is the significance of this result for the two iteration schemes?
[Hint: You may assume that the eigenvalues of the matrix in are
with corresponding eigenvectors
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Paper 3, Section II, K
Part II, 2016 commentConsider the system in scalar variables, for :
where is given, are observed at , but and are unobservable, and are independent random variables with mean 0 and variance . Define to be the estimator of with minimum variance amongst all estimators that are unbiased and linear functions of . Suppose and its variance is . After observation at of , a new unbiased estimator of , linear in , is expressed
Find and to minimize the variance of . Hence find in terms of , and . Calculate .
Suppose are Gaussian and thus . Consider minimizing , under the constraint that the control can only depend on . Show that the value function of dynamic programming for this problem can be expressed
where and is independent of and linear in .
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Paper 4, Section II, K
Part II, 2016 commentState transversality conditions that can be used with Pontryagin's maximum principle and say when they are helpful.
Given , it is desired to maximize , where
and is a time-varying control such that and . Suppose that and are positive, and that and . Find the optimal control at times close to . Show that over the optimal control is constant, or makes exactly one switch, the latter happening if and only if
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Paper 2, Section II, K
Part II, 2016 commentConsider a Markov decision problem with finite state space , value function and dynamic programming equation , where
Suppose , and for all . Prove there exists a deterministic stationary Markov policy that is optimal, explaining what the italicised words mean.
Let , where , and . Prove that
Deduce that the value iteration algorithm converges to an optimal policy in a finite number of iterations.
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Paper 1, Section II, A
Part II, 2016 commentA particle in one dimension has position and momentum operators and whose eigenstates obey
For a state , define the position-space and momentum-space wavefunctions and and show how each of these can be expressed in terms of the other.
Write down the translation operator and check that your expression is consistent with the property . For a state , relate the position-space and momentum-space wavefunctions for to and respectively.
Now consider a harmonic oscillator with mass , frequency , and annihilation and creation operators
Let and be the wavefunctions corresponding to the normalised energy eigenstates , where .
(i) Express explicitly in terms of the wavefunctions .
(ii) Given that , where the are polynomials and , show that
[You may quote standard results for a harmonic oscillator. You may also use, without proof, for operators and which each commute with
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Paper 3, Section II, 31 A
Part II, 2016 commentA three-dimensional oscillator has Hamiltonian
where the constants are real and positive. Assuming a unique ground state, construct the general normalised eigenstate of and give a formula for its energy eigenvalue. [You may quote without proof results for a one-dimensional harmonic oscillator of mass and frequency that follow from writing and
List all states in the four lowest energy levels of in the cases:
(i) ;
(ii) and , where .
Now consider with subject to a perturbation
where is small. Compute the changes in energies for the ground state and the states at the first excited level of the original Hamiltonian, working to the leading order at which nonzero corrections occur. [You may quote without proof results from perturbation theory.]
Explain briefly why some energy levels of the perturbed Hamiltonian will be exactly degenerate. [Hint: Compare with (ii) above.]
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Paper 4, Section II, A
Part II, 2016 comment(a) Consider a quantum system with Hamiltonian , where is independent of time. Define the interaction picture corresponding to this Hamiltonian and derive an expression for the time derivative of an operator in the interaction picture, assuming it is independent of time in the Schrödinger picture.
(b) The Pauli matrices satisfy
Explain briefly how these properties allow to be used to describe a quantum system with spin .
(c) A particle with spin has position and momentum operators and . The unitary operator corresponding to a rotation through an angle about an axis is where is the total angular momentum. Check this statement by considering the effect of an infinitesimal rotation on and .
(d) Suppose that the particle in part (c) has Hamiltonian with
where is the orbital angular momentum and are constants. Show that all components of are independent of time in the interaction picture. Is this true in the Heisenberg picture?
[You may quote commutation relations of with and .]
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Paper 2, Section II, A
Part II, 2016 comment(a) Let be standard, normalised angular momentum eigenstates with labels specifying eigenvalues for and . Taking units in which ,
Check the coefficients above by computing norms of states, quoting any angular momentum commutation relations that you require.
(b) Two particles, each of spin , have combined spin states . Find expressions for all such states with in terms of product states.
(c) Suppose that the particles in part (b) move about their centre of mass with a spatial wavefunction that is a spherically symmetric function of relative position. If the particles are identical, what spin states are allowed? Justify your answer.
(d) Now consider two particles of spin 1 that are not identical and are both at rest. If the 3-component of the spin of each particle is zero, what is the probability that their total, combined spin is zero?
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Paper 3, Section II, J
Part II, 2016 commentLet be i.i.d. random variables from a distribution, , and consider a Bayesian model for the unknown parameter, where is a fixed constant.
(a) Derive the posterior distribution of .
(b) Construct a credible set such that
(i) for every , and
(ii) for any ,
where denotes the distribution of the infinite sequence when drawn independently from a fixed distribution.
[You may use the central limit theorem.]
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Paper 2, Section II,
Part II, 2016 comment(a) State and prove the Cramér-Rao inequality in a parametric model , where . [Necessary regularity conditions on the model need not be specified.]
(b) Let be i.i.d. Poisson random variables with unknown parameter . For and define
Show that for all values of .
Now suppose is an estimator of with possibly nonzero bias . Suppose the function is monotone increasing on . Prove that the mean-squared errors satisfy
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Paper 4, Section II, J
Part II, 2016 commentConsider a decision problem with parameter space . Define the concepts of a Bayes decision rule and of a least favourable prior.
Suppose is a prior distribution on such that the Bayes risk of the Bayes rule equals , where is the risk function associated to the decision problem. Prove that is least favourable.
Now consider a random variable arising from the binomial distribution , where . Construct a least favourable prior for the squared risk . [You may use without proof the fact that the Bayes rule for quadratic risk is given by the posterior mean.]
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Paper 1, Section II,
Part II, 2016 commentDerive the maximum likelihood estimator based on independent observations that are identically distributed as , where the unknown parameter lies in the parameter space . Find the limiting distribution of as .
Now define
and find the limiting distribution of as .
Calculate
for the choices and . Based on the above findings, which estimator of would you prefer? Explain your answer.
[Throughout, you may use standard facts of stochastic convergence, such as the central limit theorem, provided they are clearly stated.]
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Paper 3, Section II, J
Part II, 2016 comment(a) Define the Borel -algebra and the Borel functions.
(b) Give an example with proof of a set in which is not Lebesgue measurable.
(c) The Cantor set is given by
(i) Explain why is Lebesgue measurable.
(ii) Compute the Lebesgue measure of .
(iii) Is every subset of Lebesgue measurable?
(iv) Let be the function given by
Explain why is a Borel function.
(v) Using the previous parts, prove the existence of a Lebesgue measurable set which is not Borel.
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Paper 4, Section II, J
Part II, 2016 commentGive the definitions of the convolution and of the Fourier transform of , and show that . State what it means for Fourier inversion to hold for a function .
State the Plancherel identity and compute the norm of the Fourier transform of the function .
Suppose that are functions in such that in as . Show that uniformly.
Give the definition of weak convergence, and state and prove the Central Limit Theorem.
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Paper 2, Section II, J
Part II, 2016 comment(a) State Jensen's inequality. Give the definition of and the space for . If , is it true that ? Justify your answer. State and prove Hölder's inequality using Jensen's inequality.
(b) Suppose that is a finite measure space. Show that if and then . Give the definition of and show that as .
(c) Suppose that . Show that if belongs to both and , then for any . If , must we have ? Give a proof or a counterexample.
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Paper 1, Section II, J
Part II, 2016 commentThroughout this question is a measure space and are measurable functions.
(a) Give the definitions of pointwise convergence, pointwise a.e. convergence, and convergence in measure.
(b) If pointwise a.e., does in measure? Give a proof or a counterexample.
(c) If in measure, does pointwise a.e.? Give a proof or a counterexample.
(d) Now suppose that and that is Lebesgue measure on . Suppose is a sequence of Borel measurable functions on which converges pointwise a.e. to .
(i) For each let . Show that for each .
(ii) Show that for every there exists a set with so that uniformly on .
(iii) Does (ii) hold with replaced by ? Give a proof or a counterexample.
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Paper 3, Section II, I
Part II, 2016 comment(a) Let the finite group act on a finite set and let be the permutation character. If is 2 -transitive on , show that , where is an irreducible character of .
(b) Let , and let be the symmetric group acting naturally on the set . For any integer , write for the set of all -element subsets of , and let be the permutation character of the action of on . Compute the degree of . If , compute the character inner product .
Let if is even, and if is odd. Deduce that has distinct irreducible characters such that for all ,
(c) Let be the set of all ordered pairs with and . Let act on in the obvious way. Write for the permutation character of in this action. By considering inner products, or otherwise, prove that
where is an irreducible character. Calculate the degree of , and calculate its value on the elements and of .
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Paper 2, Section II, I
Part II, 2016 commentShow that the 1-dimensional (complex) characters of a finite group form a group under pointwise multiplication. Denote this group by . Show that if , the map from to is a character of , hence an element of . What is the kernel of the ?
Show that if is abelian the map is an isomorphism. Deduce, from the structure theorem for finite abelian groups, that the groups and are isomorphic as abstract groups.
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Paper 4, Section II, I
Part II, 2016 commentLet be a proper normal subgroup of a finite group and let be an irreducible complex representation of . Show that either restricted to is a sum of copies of a single irreducible representation of , or else is induced from an irreducible representation of some proper subgroup of .
Recall that a -group is a group whose order is a power of the prime number . Deduce, by induction on the order of the group, or otherwise, that every irreducible complex representation of a -group is induced from a 1-dimensional representation of some subgroup.
[You may assume that a non-abelian -group has an abelian normal subgroup which is not contained in the centre of .]
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Paper 1, Section II, I
Part II, 2016 commentLet be a normal subgroup of the finite group . Explain how a (complex) representation of gives rise to an associated representation of , and briefly describe which representations of arise this way.
Let be the group of order 54 which is given by
Find the conjugacy classes of . By observing that and are normal in , or otherwise, construct the character table of .
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Paper 3, Section II, H
Part II, 2016 commentLet be a non-constant elliptic function with respect to a lattice . Let be a fundamental parallelogram and let the degree of be . Let denote the zeros of in , and let denote the poles (both with possible repeats). By considering the integral (if required, also slightly perturbing )
show that
Let denote the Weierstrass -function with respect to . For with we set
an elliptic function with periods . Suppose and . Prove that if and only if . [You may use standard properties of the Weierstrass -function provided they are clearly stated.]
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Paper 2, Section II, H
Part II, 2016 commentSuppose that is a holomorphic map of complex tori, and let denote the projection map for . Show that there is a holomorphic map such that
Prove that for some . Hence deduce that two complex tori and are conformally equivalent if and only if the lattices are related by for some .
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Paper 1, Section II, H
Part II, 2016 comment(a) Let be a non-constant holomorphic map between Riemann surfaces. Prove that takes open sets of to open sets of .
(b) Let be a simply connected domain strictly contained in . Is there a conformal equivalence between and ? Justify your answer.
(c) Let be a compact Riemann surface and a discrete subset. Given a non-constant holomorphic function , show that is dense in .
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Paper 2, Section I, K
Part II, 2016 commentDefine an exponential dispersion family. Prove that the range of the natural parameter, , is an open interval. Derive the mean and variance as a function of the log normalizing constant.
[Hint: Use the convexity of , i.e. for all
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Paper 4, Section I, K
Part II, 2016 comment(a) Let where for are independent and identically distributed. Let for , and suppose that these variables follow a binary regression model with the complementary log-log link function . What is the probability density function of ?
(b) The Newton-Raphson algorithm can be applied to compute the MLE, , in certain GLMs. Starting from , we let be the maximizer of the quadratic approximation of the log-likelihood around :
where and are the gradient and Hessian of the log-likelihood. What is the difference between this algorithm and Iterative Weighted Least Squares? Why might the latter be preferable?
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Paper 3, Section I, K
Part II, 2016 commentThe command
rainfall month+elnino+month:elnino)
performs a Box-Cox transform of the response at several values of the parameter , and produces the following plot:

We fit two linear models and obtain the Q-Q plots for each fit, which are shown below in no particular order:


Define the variable on the -axis in the output of boxcox, and match each Q-Q plot to one of the models.
After choosing the model fit.2, the researcher calculates Cook's distance for the th sample, which has high leverage, and compares it to the upper -point of an distribution, because the design matrix is of size . Provide an interpretation of this comparison in terms of confidence sets for . Is this confidence statement exact?
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Paper 1, Section I, K
Part II, 2016 commentThe body mass index (BMI) of your closest friend is a good predictor of your own BMI. A scientist applies polynomial regression to understand the relationship between these two variables among 200 students in a sixth form college. The commands
fit. poly friendBMI , 2, raw=T
fit. poly friendBMI, 3, raw
fit the models and , respectively, with in each case.
Setting the parameters raw to FALSE:
fit. poly friendBMI , 2, raw=F )
fit. poly friendBMI, 3, raw
fits the models and , with . The function is a polynomial of degree . Furthermore, the design matrix output by the function poly with raw=F satisfies:
poly friendBMI, 3, raw poly , raw
How does the variance of differ in the models and ? What about the variance of the fitted values ? Finally, consider the output of the commands
(fit.1,fit.2)
anova(fit.3,fit.4)
Define the test statistic computed by this function and specify its distribution. Which command yields a higher statistic?
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Paper 4, Section II, K
Part II, 2016 commentFor 31 days after the outbreak of the 2014 Ebola epidemic, the World Health Organization recorded the number of new cases per day in 60 hospitals in West Africa. Researchers are interested in modelling , the number of new Ebola cases in hospital on day , as a function of several covariates:
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lab: a Boolean factor for whether the hospital has laboratory facilities,
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casesBefore: number of cases at the hospital on the previous day,
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urban: a Boolean factor indicating an urban area,
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country: a factor with three categories, Guinea, Liberia, and Sierra Leone,
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numDoctors: number of doctors at the hospital,
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tradBurials: a Boolean factor indicating whether traditional burials are common in the region.
Consider the output of the following code (with some lines omitted):
fit. 1 <- glm(newCases lab+casesBefore+urban+country+numDoctors+tradBurials,
- data=ebola, family=poisson)
summary (fit.1)
Coefficients:
Estimate Std. Error z value
casesBefore
countryLiberia
countrySierra Leone
numDoctors
tradBurialstrUE
Signif. codes:
(a) Would you conclude based on the -tests that an urban setting does not affect the rate of infection?
(b) Explain how you would predict the total number of new cases that the researchers will record in Sierra Leone on day 32 .
We fit a new model which includes an interaction term, and compute a test statistic using the code:
fit. glm (newCases casesBefore+country+country:casesBefore+numDoctors,
- data=ebola, family=poisson)
fit. 2 deviance - fit.1$deviance
[1]
(c) What is the distribution of the statistic computed in the last line?
(d) Under what conditions is the deviance of each model approximately chi-squared?
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Paper 1, Section II, K
Part II, 2016 comment(a) Let be an -vector of responses from the linear model , with . The internally studentized residual is defined by
where is the least squares estimate, is the leverage of sample , and
Prove that the joint distribution of is the same in the following two models: (i) , and (ii) , with (in this model, are identically -distributed). [Hint: A random vector is spherically symmetric if for any orthogonal matrix . If is spherically symmetric and a.s. nonzero, then is a uniform point on the sphere; in addition, any orthogonal projection of is also spherically symmetric. A standard normal vector is spherically symmetric.]
(b) A social scientist regresses the income of 120 Cambridge graduates onto 20 answers from a questionnaire given to the participants in their first year. She notices one questionnaire with very unusual answers, which she suspects was due to miscoding. The sample has a leverage of . To check whether this sample is an outlier, she computes its externally studentized residual,
where is estimated from a fit of all samples except the one in question, . Is this a high leverage point? Can she conclude this sample is an outlier at a significance level of ?
(c) After examining the following plot of residuals against the response, the investigator calculates the externally studentized residual of the participant denoted by the black dot, which is . Can she conclude this sample is an outlier with a significance level of ?

Part II, List of Questions
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Paper 1, Section II, C
Part II, 2016 commentConsider an ideal quantum gas with one-particle states of energy . Let denote the probability that state is occupied by particles. Here, can take the values 0 or 1 for fermions and any non-negative integer for bosons. The entropy of the gas is given by
(a) Write down the constraints that must be satisfied by the probabilities if the average energy and average particle number are kept at fixed values.
Show that if is maximised then
where and are Lagrange multipliers. What is ?
(b) Insert these probabilities into the expression for , and combine the result with the first law of thermodynamics to find the meaning of and .
(c) Calculate the average occupation number for a gas of fermions.
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Paper 3, Section II, C
Part II, 2016 comment(a) Consider an ideal gas consisting of identical classical particles of mass moving freely in a volume with Hamiltonian . Show that the partition function of the gas has the form
and find as a function of the temperature .
[You may assume that for .]
(b) A monatomic gas of interacting particles is a modification of an ideal gas where any pair of particles with separation interact through a potential energy . The partition function for this gas can be written as
where . The virial expansion of the equation of state for small densities is
Using the free energy, show that
(c) The Lennard-Jones potential is
where and are positive constants. Find the separation where and the separation where has its minimum. Sketch the graph of . Calculate for this potential using the approximations
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Paper 4, Section II, 33C
Part II, 2016 comment(a) State the first law of thermodynamics. Derive the Maxwell relation
(b) Consider a thermodynamic system whose energy at constant temperature is volume independent, i.e.
Show that this implies that the pressure has the form for some function .
(c) For a photon gas inside a cavity of volume , the energy and pressure are given in terms of the energy density , which depends only on the temperature , by
Show that this implies where is a constant. Show that the entropy is
and calculate the energy and free energy in terms of their respective fundamental variables.
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Paper 2, Section II, C
Part II, 2016 comment(a) What is meant by the canonical ensemble? Consider a system in the canonical ensemble that can be in states with energies . Write down the partition function for this system and the probability that the system is in state . Derive an expression for the average energy in terms of the partition function.
(b) Consider an anharmonic oscillator with energy levels
where is a positive constant and is a small constant. Let the oscillator be in contact with a reservoir at temperature . Show that, to linear order in , the partition function for the oscillator is given by
where and are constants to be determined. Also show that, to linear order in , the average energy of a system of uncoupled oscillators of this type is given by
where are constants to be determined.
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Paper 3, Section II, K
Part II, 2016 commentConsider the following two-period market model. There is a risk-free asset which pays interest at rate . There is also a risky stock with prices given by

The above diagram should be read as
and so forth.
(a) Find the risk-neutral probabilities.
(b) Consider a European put option with strike expiring at time . What is the initial no-arbitrage price of the option? How many shares should be held in each period to replicate the payout?
(c) Now consider an American put option with the same strike and expiration date. Find the optimal exercise policy, assuming immediate exercise is not allowed. Would your answer change if you were allowed to exercise the option at time 0 ?
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Paper 4, Section II, K
Part II, 2016 commentLet be concave and strictly increasing, and let be a vector space of random variables. For every random variable let
and suppose there exists a random variable such that
For a random variable , let be such that .
(a) Show that for every constant we have , and that if , then . Hence show that if for constants , then
(b) Show that is concave, and hence show is decreasing for .
(c) Assuming is continuously differentiable, show that converges as , and that there exists a random variable such that
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Paper 2, Section II, K
Part II, 2016 commentIn the context of the Black-Scholes model, let be the initial price of the stock, and let be its volatility. Assume that the risk-free interest rate is zero and the stock pays no dividends. Let denote the initial price of a European call option with strike and maturity date .
(a) Show that the Black-Scholes formula can be written in the form
where and depend on and , and is the standard normal distribution function.
(b) Let be the initial price of a put option with strike and maturity . Show that
(c) Show that
(d) Consider a European contingent claim with maturity and payout
Assuming , show that its initial price can be written as for a volatility parameter which you should express in terms of and .
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Paper 1, Section II, 28K
Part II, 2016 comment(a) What is a Brownian motion?
(b) State the Brownian reflection principle. State the Cameron-Martin theorem for Brownian motion with constant drift.
(c) Let be a Brownian motion. Show that
where is the standard normal distribution function.
(d) Find
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Paper 1, Section I, H
Part II, 2016 commentBy considering the function defined by
or otherwise, show that there exist and such that
for all .
Show, quoting carefully any theorems you use, that we must have as .
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Paper 2, Section I, H
Part II, 2016 commentDefine what it means for a subset of to be convex. Which of the following statements about a convex set in (with the usual norm) are always true, and which are sometimes false? Give proofs or counterexamples as appropriate.
(i) The closure of is convex.
(ii) The interior of is convex.
(iii) If is linear, then is convex.
(iv) If is continuous, then is convex.
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Paper 3, Section I,
Part II, 2016 commentIn the game of 'Chicken', and drive fast cars directly at each other. If they both swerve, they both lose 10 status points; if neither swerves, they both lose 100 status points. If one swerves and the other does not, the swerver loses 20 status points and the non-swerver gains 40 status points. Find all the pairs of probabilistic strategies such that, if one driver maintains their strategy, it is not in the interest of the other to change theirs.
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Paper 4, Section I, H
Part II, 2016 commentLet be integers such that there exists an with for all . Show that, if infinitely many of the are non-zero, then is an irrational number.
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Paper 2, Section II, H
Part II, 2016 commentProve Bernstein's theorem, which states that if is continuous and
then uniformly on . [Theorems from probability theory may be used without proof provided they are clearly stated.]
Deduce Weierstrass's theorem on polynomial approximation for any closed interval.
Proving any results on Chebyshev polynomials that you need, show that, if is continuous and , then we can find an and , for , such that
for all . Deduce that as .
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Paper 4, Section II, H
Part II, 2016 commentExplain briefly how a positive irrational number gives rise to a continued fraction
with the non-negative integers and for .
Show that, if we write
then
for .
Use the observation [which need not be proved] that lies between and to show that
Show that where is the th Fibonacci number (thus , , and conclude that
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Paper 4, Section II, D
Part II, 2016 commentA duck swims at a constant velocity , where , on the surface of infinitely deep water. Surface tension can be neglected, and the dispersion relation for the linear surface water waves (relative to fluid at rest) is . Show that the wavevector of a plane harmonic wave that is steady in the duck's frame, i.e. of the form
where and are horizontal coordinates relative to the duck, satisfies
where and . [You may assume that ]
Assume that the wave pattern behind the duck can be regarded as a Fourier superposition of such steady waves, i.e., the surface elevation at has the form
where
Show that, in the limit at fixed with ,
where
and denotes . Briefly interpret this result in terms of what is seen.
Without doing detailed calculations, briefly explain what is seen as at fixed with . Very briefly comment on the case .
[Hint: You may find the following results useful.
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Paper 2, Section II, 37D
Part II, 2016 commentStarting from the equations for one-dimensional unsteady flow of a perfect gas at constant entropy, show that the Riemann invariants
are constant on characteristics given by , where is the speed of the gas, is the local speed of sound, is a constant and is the exponent in the adiabatic equation of state for .
At time the gas occupies and is at rest at uniform density , pressure and sound speed . For , a piston initially at has position , where
and and are positive constants. For the case , sketch the piston path and the characteristics in in the -plane, and find the time and place at which a shock first forms in the gas.
Do likewise for the case .
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Paper 1, Section II, D
Part II, 2016 commentWrite down the linearised equations governing motion of an inviscid compressible fluid at uniform entropy. Assuming that the velocity is irrotational, show that it may be derived from a velocity potential satisfying the wave equation
and identify the wave speed . Obtain from these linearised equations the energyconservation equation
and give expressions for the acoustic-energy density and the acoustic-energy flux in terms of .
Such a fluid occupies a semi-infinite waveguide of square cross-section , bounded by rigid walls. An impenetrable membrane closing the end makes prescribed small displacements to
where and . Show that the velocity potential is given by
where the functions , including their amplitudes, are to be determined, with the sign of any square roots specified clearly.
If , what is the asymptotic behaviour of as ? Using this behaviour and the energy-conservation equation averaged over both time and the crosssection, or otherwise, determine the double-averaged energy flux along the waveguide,
explaining why this is independent of .
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Paper 3, Section II,
Part II, 2016 commentSmall disturbances in a homogeneous elastic solid with density and Lamé moduli and are governed by the equation
where is the displacement. Show that a harmonic plane-wave solution
must satisfy
where the wavespeeds and are to be identified. Describe mathematically how such plane-wave solutions can be classified into longitudinal -waves and transverse - and -waves (taking the -direction as the vertical direction).
The half-space is filled with the elastic solid described above, while the slab is filled with a homogeneous elastic solid with Lamé moduli and , and wavespeeds and . There is a rigid boundary at . A harmonic plane -wave propagates from towards the interface , with displacement
How are and related? The total displacement in is the sum of and that of the reflected -wave,
Write down the form of the displacement in , and determine the (complex) reflection coefficient . Verify that regardless of the parameter values, and explain this physically.
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Paper 3, Section II,
Part II, 2016 commentSmall disturbances in a homogeneous elastic solid with density and Lamé moduli and are governed by the equation
where is the displacement. Show that a harmonic plane-wave solution
must satisfy
where the wavespeeds and are to be identified. Describe mathematically how such plane-wave solutions can be classified into longitudinal -waves and transverse - and -waves (taking the -direction as the vertical direction).
The half-space is filled with the elastic solid described above, while the slab is filled with a homogeneous elastic solid with Lamé moduli and , and wavespeeds and . There is a rigid boundary at . A harmonic plane -wave propagates from towards the interface , with displacement
How are and related? The total displacement in is the sum of and that of the reflected -wave,
Write down the form of the displacement in , and determine the (complex) reflection coefficient . Verify that regardless of the parameter values, and explain this physically.